TPTP Problem File: SLH0516^1.p
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%------------------------------------------------------------------------------
% 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_01479_047243__6009312_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1377 ( 535 unt; 102 typ; 0 def)
% Number of atoms : 3775 (1229 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11812 ( 311 ~; 109 |; 217 &;9461 @)
% ( 0 <=>;1714 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 585 ( 585 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 94 usr; 13 con; 0-3 aty)
% Number of variables : 3722 ( 276 ^;3344 !; 102 ?;3722 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:58.154
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_Ri1641125681238393385ccount: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (94)
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Risk____Free____Lending__Oaccount,type,
minus_4846202936726426316ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__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__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__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_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
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__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__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_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
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_If_001t__Risk____Free____Lending__Oaccount,type,
if_Risk_Free_account: $o > risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
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__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_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_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
ord_le5106303358561053821ccount: set_Ri1641125681238393385ccount > set_Ri1641125681238393385ccount > $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__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_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $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_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_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_Oreturn__loans,type,
risk_F2121631595377017831_loans: ( nat > real ) > risk_Free_account > 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_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
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_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Risk____Free____Lending__Oaccount,type,
set_or4484699493994522366ccount: risk_Free_account > risk_Free_account > set_Ri1641125681238393385ccount ).
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_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $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__092_060alpha_062,type,
alpha: risk_Free_account ).
thf(sy_v__092_060alpha_062_H____,type,
alpha2: risk_Free_account ).
thf(sy_v__092_060beta_062,type,
beta: risk_Free_account ).
thf(sy_v__092_060rho_062,type,
rho: nat > real ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1265)
thf(fact_0_assms_I1_J,axiom,
! [N: nat] : ( ord_less_real @ ( rho @ N ) @ one_one_real ) ).
% assms(1)
thf(fact_1__092_060open_0620_A_060_A1_A_N_A_092_060rho_062_An_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ ( minus_minus_real @ one_one_real @ ( rho @ n ) ) ).
% \<open>0 < 1 - \<rho> n\<close>
thf(fact_2__092_060open_0620_A_092_060le_062_Asum_A_I_092_060pi_062_A_092_060alpha_062_J_A_123_O_On_125_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ alpha2 ) @ ( set_ord_atMost_nat @ n ) ) ).
% \<open>0 \<le> sum (\<pi> \<alpha>) {..n}\<close>
thf(fact_3__092_060open_062_I1_A_N_A_092_060rho_062_An_J_A_K_Asum_A_I_092_060pi_062_A_092_060alpha_062_J_A_123_O_On_125_A_092_060le_062_A_I_092_060Sum_062i_092_060le_062n_O_A_I1_A_N_A_092_060rho_062_Ai_J_A_K_A_092_060pi_062_A_092_060alpha_062_Ai_J_092_060close_062,axiom,
( ord_less_eq_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ ( rho @ n ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ alpha2 ) @ ( set_ord_atMost_nat @ n ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( rho @ I ) ) @ ( risk_F170160801229183585ccount @ alpha2 @ I ) )
@ ( set_ord_atMost_nat @ n ) ) ) ).
% \<open>(1 - \<rho> n) * sum (\<pi> \<alpha>) {..n} \<le> (\<Sum>i\<le>n. (1 - \<rho> i) * \<pi> \<alpha> i)\<close>
thf(fact_4__092_060open_062_092_060forall_062n_O_A0_A_092_060le_062_Asum_A_I_092_060pi_062_A_092_060alpha_062_J_A_123_O_On_125_092_060close_062,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ alpha2 ) @ ( set_ord_atMost_nat @ N ) ) ) ).
% \<open>\<forall>n. 0 \<le> sum (\<pi> \<alpha>) {..n}\<close>
thf(fact_5_assms_I2_J,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_real @ ( rho @ N ) @ ( rho @ M ) ) ) ).
% assms(2)
thf(fact_6_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_7_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_8_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_9_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_10_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_11_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_12_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_13_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_14_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_15_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_16_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_17_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_18_diff__ge__0__iff__ge,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( minus_4846202936726426316ccount @ A @ B ) )
= ( ord_le4245800335709223507ccount @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_19_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_20_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_21_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_22_atMost__iff,axiom,
! [I2: risk_Free_account,K: risk_Free_account] :
( ( member5612106785598075018ccount @ I2 @ ( set_or3854930313887350124ccount @ K ) )
= ( ord_le4245800335709223507ccount @ I2 @ K ) ) ).
% atMost_iff
thf(fact_23_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_24_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_25_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_26_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_27_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_28_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_29_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_30_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_31_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_32_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_33_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_34_diff__self,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ A )
= zero_z1425366712893667068ccount ) ).
% diff_self
thf(fact_35_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_36__092_060open_0620_A_092_060le_062_A_092_060alpha_062_092_060close_062,axiom,
ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ alpha2 ).
% \<open>0 \<le> \<alpha>\<close>
thf(fact_37_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_38_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_39_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_40_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_41_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_42_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_43_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_44_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_45_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_46_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_47_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_48_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_49_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_50_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_51_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_52_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_53_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_54_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_55_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_56_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_57_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ A )
= zero_z1425366712893667068ccount ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_58_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_59_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_60_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_61_diff__zero,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_zero
thf(fact_62_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_63_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_64_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_65_diff__0__right,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_0_right
thf(fact_66_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_67_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_68_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_69_diff__gt__0__iff__gt,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( minus_4846202936726426316ccount @ A @ B ) )
= ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_70_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_71_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_72_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 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_73_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_74_gr__implies__not__zero,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_75_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_76_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_77_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_78_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_79_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_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_84_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_85_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_86_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_87_diff__strict__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_88_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_89_diff__strict__left__mono,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ C @ A ) @ ( minus_4846202936726426316ccount @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_90_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_91_diff__eq__diff__less,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( ord_le2131251472502387783ccount @ A @ B )
= ( ord_le2131251472502387783ccount @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_92_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_93_diff__strict__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,D: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ D @ C )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_94_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_95_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_96_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_97_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_98_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_99_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_100_mult_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_101_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_102_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_103_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_104_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_105_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_106_mult_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.assoc
thf(fact_107_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_108_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_109_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_110_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_111_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_112_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_113_diff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_114_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_115_diff__right__commute,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ B )
= ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_116_diff__right__commute,axiom,
! [A: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B )
= ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_117_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_118_diff__eq__diff__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_119_diff__eq__diff__eq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_120_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_121_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: complex > real,J: real > complex,T: set_complex,H: complex > complex,G: real > complex] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_complex @ ( J @ A4 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_122_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_123_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_124_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: real > complex,J: complex > real,T: set_real,H: real > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( member_real @ ( J @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_complex @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups5754745047067104278omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_125_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: complex > complex,J: complex > complex,T: set_complex,H: complex > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( member_complex @ ( J @ A4 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_complex @ ( I2 @ B3 ) @ S ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_126_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_127_sum_Oeq__general__inverses,axiom,
! [B4: set_complex,K: complex > real,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B4 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_128_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_129_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_130_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > complex,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ( ( member_complex @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_131_sum_Oeq__general__inverses,axiom,
! [B4: set_complex,K: complex > complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B4 )
=> ( ( member_complex @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_132_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_133_sum_Oeq__general,axiom,
! [B4: set_complex,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_134_sum_Oeq__general,axiom,
! [B4: set_real,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_135_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_136_sum_Oeq__general,axiom,
! [B4: set_real,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B4 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_137_sum_Oeq__general,axiom,
! [B4: set_complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B4 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_138_sum_Ocong,axiom,
! [A2: set_nat,B4: set_nat,G: nat > real,H: nat > real] :
( ( A2 = B4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_139_sum_Ocong,axiom,
! [A2: set_complex,B4: set_complex,G: complex > complex,H: complex > complex] :
( ( A2 = B4 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= ( groups7754918857620584856omplex @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_140_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_141_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_142_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_143_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_144_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_145_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_146_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_147_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_148_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_149_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_150_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_151_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_152_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_153_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_154_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_155_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_156_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_157_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_158_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_159_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_160_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_161_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_162_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_163_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_164_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_165_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_166_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_167_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_168_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_169_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_170_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_171_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_172_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_173_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_174_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_175_less__iff__diff__less__0,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A3 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% less_iff_diff_less_0
thf(fact_176_less__1__mult,axiom,
! [M4: real,N2: real] :
( ( ord_less_real @ one_one_real @ M4 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_177_less__1__mult,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M4 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_178_sum_Oswap,axiom,
! [G: nat > nat > real,B4: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B4 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J2 )
@ A2 )
@ B4 ) ) ).
% sum.swap
thf(fact_179_sum_Oswap,axiom,
! [G: complex > complex > complex,B4: set_complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ B4 )
@ A2 )
= ( groups7754918857620584856omplex
@ ^ [J2: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J2 )
@ A2 )
@ B4 ) ) ).
% sum.swap
thf(fact_180_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N3 ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N3 ) ) ) ) ) ).
% less_eq_account_def
thf(fact_181_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_182_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_183_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_184_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_185_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_186_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_187_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_188_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_189_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_190_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_191_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_192_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_193_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_194_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_195_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_196_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_197_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_198_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_199_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_200_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_201_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_202_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_203_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_204_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_205_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_206_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_207_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_208_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_209_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_210_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_211_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_212_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_213_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_214_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_215_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_216_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_217_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_218_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_219_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_220_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_221_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_222_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_223_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_224_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_225_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_226_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_227_diff__eq__diff__less__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
= ( ord_le4245800335709223507ccount @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_228_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_229_diff__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_230_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_231_diff__left__mono,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ C @ A ) @ ( minus_4846202936726426316ccount @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_232_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_233_diff__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,D: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ D @ C )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_234_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_235_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: risk_Free_account,Z: risk_Free_account] : ( Y3 = Z ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ B2 )
= zero_z1425366712893667068ccount ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_236_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: complex,Z: complex] : ( Y3 = Z ) )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_237_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_238_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_239_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_240_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_241_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_242_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_243_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_244_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_245_right__diff__distrib_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_246_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_247_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_248_left__diff__distrib_H,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
= ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_249_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_250_right__diff__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_251_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_252_left__diff__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_253_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_254_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A2: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A2 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_255_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_256_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_257_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_258_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_259_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_260_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_261_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_262_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_263_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_264_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_265_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_266_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_267_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_268_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_269_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_270_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_271_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_272_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_273_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_274_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_275_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_276_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_277_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_278_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_279_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B4: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B4 ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_280_sum__product,axiom,
! [F: complex > complex,A2: set_complex,G: complex > complex,B4: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B4 ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J2: complex] : ( times_times_complex @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_281_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_282_sum__distrib__right,axiom,
! [F: complex > complex,A2: set_complex,R: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_283_sum__distrib__left,axiom,
! [R: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( times_times_real @ R @ ( F @ N3 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_284_sum__distrib__left,axiom,
! [R: complex,F: complex > complex,A2: set_complex] :
( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ R @ ( F @ N3 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_285_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_286_sum__subtractf,axiom,
! [F: complex > complex,G: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X3: complex] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_287_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_288_atMost__def,axiom,
( set_or3854930313887350124ccount
= ( ^ [U: risk_Free_account] :
( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] : ( ord_le4245800335709223507ccount @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_289_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_290_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_291_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_292_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_293_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_294_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_295_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_296_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_297_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_298_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_299_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_300_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_301_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_302_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_303_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_304_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_305_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_306_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_307_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_308_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_309_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_310_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_311_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_312_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_313_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_314_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_315_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_316_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_317_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_318_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_319_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_320_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_321_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_322_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_323_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_324_le__iff__diff__le__0,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A3 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% le_iff_diff_le_0
thf(fact_325_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_326_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_327_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_328_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_329_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_330_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_331_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_332_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_333_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_334_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_335_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_336_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_337_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_338_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_339_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_340_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_341_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B3: real,C2: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_eq_real @ X2 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D2 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_342_mult__le__cancel__iff2,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_343_mult__le__cancel__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_344_mult__less__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_345_Rep__account__return__loans,axiom,
! [Rho: nat > real,Alpha: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) )
= ( ^ [N3: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( Rho @ N3 ) ) @ ( risk_F170160801229183585ccount @ Alpha @ N3 ) ) ) ) ).
% Rep_account_return_loans
thf(fact_346_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_347_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_348_diff__is__0__eq,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_349_diff__is__0__eq_H,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_350_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_351_assms_I3_J,axiom,
ord_le4245800335709223507ccount @ alpha @ beta ).
% assms(3)
thf(fact_352_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_353_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_354_mult__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_355_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_356_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_357_mult__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M4 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_358_mult__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M4 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_359_mult__less__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_360_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_361_nat__0__less__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_362_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_363_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_364_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_365_zero__less__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M4 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% zero_less_diff
thf(fact_366_return__loans__zero,axiom,
! [Rho: nat > real] :
( ( risk_F2121631595377017831_loans @ Rho @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% return_loans_zero
thf(fact_367_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_368_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_369_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_370_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_371_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_372_diff__less,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ) ) ).
% diff_less
thf(fact_373_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_374_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_375_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_376_diff__less__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_377_gr__implies__not0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_378_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_379_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_380_diffs0__imp__equal,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M4 )
= zero_zero_nat )
=> ( M4 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_381_diff__mult__distrib,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_382_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N4 )
& ~ ( P @ M ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_383_diff__mult__distrib2,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M4 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_384_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_385_mult__eq__self__implies__10,axiom,
! [M4: nat,N2: nat] :
( ( M4
= ( times_times_nat @ M4 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_386_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_387_return__loans__subtract,axiom,
! [Rho: nat > real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2121631595377017831_loans @ Rho @ ( minus_4846202936726426316ccount @ Alpha @ Beta ) )
= ( minus_4846202936726426316ccount @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) @ ( risk_F2121631595377017831_loans @ Rho @ Beta ) ) ) ).
% return_loans_subtract
thf(fact_388_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_389_le__neq__implies__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( M4 != N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_390_less__or__eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_391_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_392_less__imp__le__nat,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_393_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_394_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_395_less__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_396_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_397_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_398_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_399_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_400_nat__le__linear,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
| ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% nat_le_linear
thf(fact_401_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_402_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_403_diff__le__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_404_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_405_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_406_diff__le__self,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ).
% diff_le_self
thf(fact_407_diff__le__mono,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_408_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_409_le__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_410_eq__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M4 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_411_le__antisym,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ( M4 = N2 ) ) ) ).
% le_antisym
thf(fact_412_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_413_eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% eq_imp_le
thf(fact_414_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_415_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_416_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_417_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_418_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_419_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_420_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_421_nat__mult__le__cancel__disj,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_422_return__loans__def,axiom,
( risk_F2121631595377017831_loans
= ( ^ [Rho2: nat > real,Alpha2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N3: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( Rho2 @ N3 ) ) @ ( risk_F170160801229183585ccount @ Alpha2 @ N3 ) ) ) ) ) ).
% return_loans_def
thf(fact_423_nat__mult__less__cancel__disj,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_424_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_425_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha2: risk_Free_account] :
! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha2 ) @ ( set_ord_atMost_nat @ N3 ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_426_nat__1__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_427_nat__mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_428_psubsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_429_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_430_nat__neq__iff,axiom,
! [M4: nat,N2: nat] :
( ( M4 != N2 )
= ( ( ord_less_nat @ M4 @ N2 )
| ( ord_less_nat @ N2 @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_431_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A5 )
@ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_432_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_433_less__not__refl2,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( M4 != N2 ) ) ).
% less_not_refl2
thf(fact_434_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_435_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_436_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( P @ M ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_437_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N4 )
& ~ ( P @ M ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_438_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_439_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_440_psubset__imp__ex__mem,axiom,
! [A2: set_real,B4: set_real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_441_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_442_in__mono,axiom,
! [A2: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_443_subsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_444_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( member_real @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_445_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A5 )
=> ( member_real @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_446_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_447_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_448_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M4 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_449_nat__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_450_nat__mult__eq__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N2 ) )
= ( M4 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_451_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_452_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_453_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_454_Collect__subset,axiom,
! [A2: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_455_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A5 )
@ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_456_nat__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_457_strictly__solvent__non__negative__cash,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_458_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups6033208628184776703ccount
@ ^ [J2: nat] : ( if_Risk_Free_account @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_Risk_Free_account @ ( J2 = K ) @ zero_z1425366712893667068ccount @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups6033208628184776703ccount
@ ^ [J2: nat] : ( if_Risk_Free_account @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_459_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_460_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > complex,H: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups2073611262835488442omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_complex @ ( J2 = K ) @ zero_zero_complex @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups2073611262835488442omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_461_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_real @ ( J2 = K ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_462_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_463_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_464_dual__order_Orefl,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% dual_order.refl
thf(fact_465_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_466_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_467_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_468_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha2: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha2 @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_469_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M4: 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 @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_470_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_471_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_472_dual__order_Otrans,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le4245800335709223507ccount @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_473_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_474_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_475_DiffI,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B4 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_476_Diff__iff,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_477_Suc__less__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_less_eq
thf(fact_478_Suc__mono,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_479_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_480_Suc__le__mono,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% Suc_le_mono
thf(fact_481_diff__Suc__Suc,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_482_Suc__diff__diff,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M4 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_483_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_484_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_485_mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_486_one__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_487_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_488_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_489_one__le__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_490_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_491_DiffE,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% DiffE
thf(fact_492_DiffD1,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_493_DiffD2,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ~ ( member_real @ C @ B4 ) ) ).
% DiffD2
thf(fact_494_set__diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B5: set_real] :
( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A5 )
& ~ ( member_real @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_495_set__diff__eq,axiom,
( minus_811609699411566653omplex
= ( ^ [A5: set_complex,B5: set_complex] :
( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A5 )
& ~ ( member_complex @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_496_minus__set__def,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B5: set_real] :
( collect_real
@ ( minus_minus_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A5 )
@ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_497_minus__set__def,axiom,
( minus_811609699411566653omplex
= ( ^ [A5: set_complex,B5: set_complex] :
( collect_complex
@ ( minus_8727706125548526216plex_o
@ ^ [X3: complex] : ( member_complex @ X3 @ A5 )
@ ^ [X3: complex] : ( member_complex @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_498_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_499_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_500_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_501_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_502_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_503_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_504_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_505_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_506_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M4 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_507_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_508_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_509_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_510_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_511_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_512_not__less__less__Suc__eq,axiom,
! [N2: nat,M4: nat] :
( ~ ( ord_less_nat @ N2 @ M4 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
= ( N2 = M4 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_513_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_514_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [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 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_515_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_516_Suc__less__SucD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_517_less__antisym,axiom,
! [N2: nat,M4: nat] :
( ~ ( ord_less_nat @ N2 @ M4 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
=> ( M4 = N2 ) ) ) ).
% less_antisym
thf(fact_518_Suc__less__eq2,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M4 )
= ( ? [M6: nat] :
( ( M4
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_519_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_520_not__less__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M4 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M4 ) ) ) ).
% not_less_eq
thf(fact_521_less__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_522_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_523_less__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ M4 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_524_less__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M4 @ N2 )
=> ( M4 = N2 ) ) ) ).
% less_SucE
thf(fact_525_Suc__lessI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ( suc @ M4 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M4 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_526_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_527_Suc__lessD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_lessD
thf(fact_528_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_529_transitive__stepwise__le,axiom,
! [M4: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M4 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_530_nat__induct__at__least,axiom,
! [M4: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( P @ M4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_531_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N4 )
=> ( P @ M ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_532_not__less__eq__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_533_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_534_le__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M4 @ N2 )
| ( M4
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_535_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_536_le__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_537_le__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N2 )
=> ( M4
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_538_Suc__leD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_leD
thf(fact_539_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_540_Suc__mult__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M4 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M4 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_541_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_542_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_543_lift__Suc__mono__le,axiom,
! [F: nat > risk_Free_account,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_le4245800335709223507ccount @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le4245800335709223507ccount @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_544_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_545_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_546_lift__Suc__antimono__le,axiom,
! [F: nat > risk_Free_account,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_le4245800335709223507ccount @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le4245800335709223507ccount @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_547_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M4: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M4 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_548_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M4: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M4 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_549_lift__Suc__mono__less__iff,axiom,
! [F: nat > risk_Free_account,N2: nat,M4: nat] :
( ! [N4: nat] : ( ord_le2131251472502387783ccount @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_le2131251472502387783ccount @ ( F @ N2 ) @ ( F @ M4 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_550_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_551_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_552_lift__Suc__mono__less,axiom,
! [F: nat > risk_Free_account,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_le2131251472502387783ccount @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_le2131251472502387783ccount @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_553_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_554_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_555_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_556_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_557_less__Suc__eq__0__disj,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ( M4 = zero_zero_nat )
| ? [J2: nat] :
( ( M4
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_558_Suc__leI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_leI
thf(fact_559_Suc__le__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_eq
thf(fact_560_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_561_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_562_Suc__le__lessD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_563_le__less__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
= ( N2 = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_564_less__Suc__eq__le,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_565_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_566_le__imp__less__Suc,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_567_Suc__diff__Suc,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( ( suc @ ( minus_minus_nat @ M4 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_568_diff__less__Suc,axiom,
! [M4: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M4 @ N2 ) @ ( suc @ M4 ) ) ).
% diff_less_Suc
thf(fact_569_Suc__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_570_Suc__diff__le,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_571_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ ( suc @ X2 ) @ A2 )
=> ( ( F @ ( suc @ X2 ) )
= ( G @ ( suc @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_572_Suc__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_573_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_574_diff__Suc__eq__diff__pred,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ M4 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_575_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_576_diff__Suc__less,axiom,
! [N2: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_577_one__less__mult,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_578_n__less__m__mult__n,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_579_n__less__n__mult__m,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M4 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_580_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_581_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_582_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_583_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_584_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X3: real,Y5: real] :
( ( ord_less_eq_real @ X3 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_585_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_586_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z: risk_Free_account] : ( Y3 = Z ) )
= ( ^ [X3: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y5 )
& ( ord_le4245800335709223507ccount @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_587_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_588_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_589_ord__eq__le__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A = B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_590_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_591_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_592_ord__le__eq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( B = C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_593_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_594_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_595_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_596_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_597_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_598_order_Otrans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% order.trans
thf(fact_599_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_600_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_601_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_602_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_603_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_604_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_605_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_606_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z: risk_Free_account] : ( Y3 = Z ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_607_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_608_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_609_dual__order_Oantisym,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_610_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_611_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_612_antisym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_613_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_614_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_615_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z: risk_Free_account] : ( Y3 = Z ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_616_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_617_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_618_order__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_620_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_621_order__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_622_order__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_623_order__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_624_order__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_625_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_626_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_627_order__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_628_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_629_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_630_order__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_631_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_632_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_633_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_634_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_635_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_636_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_637_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_638_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_639_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_640_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_641_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_642_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_643_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_644_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_645_ord__eq__le__subst,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_646_ord__eq__le__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_647_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_648_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_649_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_650_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_651_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_652_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_653_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_654_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_655_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_656_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_657_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_658_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_659_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_660_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_661_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_662_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_663_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_664_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_665_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_666_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_667_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_668_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_669_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_670_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_681_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_682_order__less__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_683_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_684_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_685_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_686_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_687_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_688_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_689_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_690_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_691_order__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_692_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_693_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_694_order__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_695_order__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_696_order__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_697_order__less__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_698_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_699_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_700_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_701_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_702_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_703_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_704_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_705_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_706_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_707_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_708_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_711_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_712_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_713_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_714_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_715_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_716_ord__eq__less__subst,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_717_ord__eq__less__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_718_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_719_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_720_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_721_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_722_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_723_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_724_order__less__asym_H,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% order_less_asym'
thf(fact_725_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_726_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_727_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_728_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_729_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_730_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_731_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_732_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_733_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_734_dual__order_Ostrict__implies__not__eq,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_735_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_736_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_737_order_Ostrict__implies__not__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_738_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_739_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_740_dual__order_Ostrict__trans,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_741_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_742_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_743_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_744_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_745_order_Ostrict__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_746_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_747_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_748_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P4 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_749_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_750_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_751_dual__order_Oirrefl,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% dual_order.irrefl
thf(fact_752_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_753_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_754_dual__order_Oasym,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ~ ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% dual_order.asym
thf(fact_755_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_756_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_757_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_758_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_759_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_760_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_761_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_762_ord__less__eq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( B = C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_763_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_764_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_765_ord__eq__less__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A = B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_766_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_767_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_768_order_Oasym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% order.asym
thf(fact_769_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_770_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_771_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_772_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_773_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_774_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_775_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_776_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_777_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N2 )
= ( minus_minus_nat @ M4 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_778_sum__telescope,axiom,
! [F: nat > risk_Free_account,I2: nat] :
( ( groups6033208628184776703ccount
@ ^ [I: nat] : ( minus_4846202936726426316ccount @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_4846202936726426316ccount @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_779_sum__telescope,axiom,
! [F: nat > complex,I2: nat] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( minus_minus_complex @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_780_sum__telescope,axiom,
! [F: nat > real,I2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_781_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_782_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_783_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_784_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_785_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_786_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_787_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_788_nless__le,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A @ B ) )
= ( ~ ( ord_le4245800335709223507ccount @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_789_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_790_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_791_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_792_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_793_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_794_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_795_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_796_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_797_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_eq_real @ X3 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_798_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_799_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y5 )
& ~ ( ord_le4245800335709223507ccount @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_800_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_801_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_802_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_803_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_804_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_805_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_806_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_807_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_808_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_809_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_810_order_Ostrict__trans1,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_811_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_812_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_813_order_Ostrict__trans2,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_814_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_815_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_816_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ~ ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_817_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_818_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_819_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_820_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_821_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_822_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_823_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_824_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_825_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_826_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_827_dual__order_Ostrict__trans1,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_828_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_829_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_830_dual__order_Ostrict__trans2,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_831_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_832_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_833_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ~ ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_834_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_835_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_836_order_Ostrict__implies__order,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_837_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_838_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_839_dual__order_Ostrict__implies__order,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ord_le4245800335709223507ccount @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_840_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_841_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_842_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X3: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_843_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_eq_real @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_844_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_845_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_846_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_847_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_848_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_849_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_850_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_851_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_852_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_853_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_854_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_855_order__le__neq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( A != B )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_856_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_857_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_858_order__neq__le__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A != B )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_859_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_860_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_861_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_862_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_863_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_864_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_865_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_866_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_867_order__le__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_868_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_869_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_870_order__le__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_871_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_872_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_873_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_874_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_875_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_876_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_877_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_878_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_879_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_880_order__le__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_881_order__le__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_882_order__le__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_883_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_884_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_885_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_886_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_887_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_888_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_889_order__less__le__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_890_order__less__le__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_891_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_892_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_893_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_894_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_895_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_896_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_897_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_898_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_899_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_900_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_901_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_902_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_903_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_904_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_905_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_906_pred__subset__eq,axiom,
! [R2: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ R2 )
@ ^ [X3: real] : ( member_real @ X3 @ S ) )
= ( ord_less_eq_set_real @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_907_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups129246275422532515t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_real @ ( J2 = K ) @ one_one_real @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups129246275422532515t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_908_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups708209901874060359at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups708209901874060359at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_909_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > complex,H: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups6464643781859351333omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_complex @ ( J2 = K ) @ one_one_complex @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups6464643781859351333omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_910_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_911_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_912_inf__period_I2_J,axiom,
! [P: real > $o,D3: real,Q: real > $o] :
( ! [X2: real,K3: real] :
( ( P @ X2 )
= ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ( ! [X2: real,K3: real] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ! [X4: real,K4: real] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D3 ) ) )
| ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_913_inf__period_I2_J,axiom,
! [P: complex > $o,D3: complex,Q: complex > $o] :
( ! [X2: complex,K3: complex] :
( ( P @ X2 )
= ( P @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K3 @ D3 ) ) ) )
=> ( ! [X2: complex,K3: complex] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K3 @ D3 ) ) ) )
=> ! [X4: complex,K4: complex] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D3 ) ) )
| ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_914_prod_OatMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_915_prod_OatMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_916_prod_OatMost__Suc,axiom,
! [G: nat > complex,N2: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_917_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_918_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups4696554848551431203al_nat @ G @ A2 )
!= one_one_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_919_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups713298508707869441omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_920_prod__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_921_prod__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_922_prod__ge__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_923_prod__ge__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_924_prod__le__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) )
& ( ord_less_eq_real @ ( F @ X2 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_925_prod__le__1,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 @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_926_minf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_927_minf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_928_minf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_929_minf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_930_minf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_931_minf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_932_minf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_933_minf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_934_minf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_935_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_936_minf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_937_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_938_pinf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_939_pinf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_940_pinf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_941_pinf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_942_pinf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_943_pinf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_944_pinf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_945_pinf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_946_pinf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_947_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_948_pinf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_949_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_950_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_951_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_952_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_953_prod_OatMost__Suc__shift,axiom,
! [G: nat > complex,N2: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( G @ zero_zero_nat )
@ ( groups6464643781859351333omplex
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_954_pinf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_955_pinf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_956_pinf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_957_pinf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_958_minf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_eq_real @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_959_minf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_960_minf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_961_minf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_962_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_963_inf__period_I1_J,axiom,
! [P: real > $o,D3: real,Q: real > $o] :
( ! [X2: real,K3: real] :
( ( P @ X2 )
= ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ( ! [X2: real,K3: real] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ! [X4: real,K4: real] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D3 ) ) )
& ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_964_inf__period_I1_J,axiom,
! [P: complex > $o,D3: complex,Q: complex > $o] :
( ! [X2: complex,K3: complex] :
( ( P @ X2 )
= ( P @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K3 @ D3 ) ) ) )
=> ( ! [X2: complex,K3: complex] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K3 @ D3 ) ) ) )
=> ! [X4: complex,K4: complex] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D3 ) ) )
& ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_965_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha2 ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_966_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_967_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_968_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_969_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_970_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha2 @ Tau ) ) ) ) ) ).
% valid_transfer_def
thf(fact_971_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_972_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_973_verit__comp__simplify1_I2_J,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_974_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_975_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_976_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_977_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_978_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_979_verit__comp__simplify1_I1_J,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_980_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_981_partial__nav__just__cash,axiom,
! [A: real,N2: nat] :
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ A ) ) @ ( set_ord_atMost_nat @ N2 ) )
= A ) ).
% partial_nav_just_cash
thf(fact_982_sum__gp__basic,axiom,
! [X: complex,N2: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_983_sum__gp__basic,axiom,
! [X: real,N2: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_984_sum__telescope_H_H,axiom,
! [M4: nat,N2: nat,F: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K5: nat] : ( minus_4846202936726426316ccount @ ( F @ K5 ) @ ( F @ ( minus_minus_nat @ K5 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) )
= ( minus_4846202936726426316ccount @ ( F @ N2 ) @ ( F @ M4 ) ) ) ) ).
% sum_telescope''
thf(fact_985_sum__telescope_H_H,axiom,
! [M4: nat,N2: nat,F: nat > complex] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K5: nat] : ( minus_minus_complex @ ( F @ K5 ) @ ( F @ ( minus_minus_nat @ K5 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) )
= ( minus_minus_complex @ ( F @ N2 ) @ ( F @ M4 ) ) ) ) ).
% sum_telescope''
thf(fact_986_sum__telescope_H_H,axiom,
! [M4: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K5: nat] : ( minus_minus_real @ ( F @ K5 ) @ ( F @ ( minus_minus_nat @ K5 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) )
= ( minus_minus_real @ ( F @ N2 ) @ ( F @ M4 ) ) ) ) ).
% sum_telescope''
thf(fact_987_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_988_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_989_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_990_Icc__eq__Icc,axiom,
! [L: risk_Free_account,H: risk_Free_account,L2: risk_Free_account,H3: risk_Free_account] :
( ( ( set_or4484699493994522366ccount @ L @ H )
= ( set_or4484699493994522366ccount @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_le4245800335709223507ccount @ L @ H )
& ~ ( ord_le4245800335709223507ccount @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_991_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_992_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U2: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_993_atLeastAtMost__iff,axiom,
! [I2: risk_Free_account,L: risk_Free_account,U2: risk_Free_account] :
( ( member5612106785598075018ccount @ I2 @ ( set_or4484699493994522366ccount @ L @ U2 ) )
= ( ( ord_le4245800335709223507ccount @ L @ I2 )
& ( ord_le4245800335709223507ccount @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_994_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U2: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_995_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_996_atLeastatMost__subset__iff,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or4484699493994522366ccount @ A @ B ) @ ( set_or4484699493994522366ccount @ C @ D ) )
= ( ~ ( ord_le4245800335709223507ccount @ A @ B )
| ( ( ord_le4245800335709223507ccount @ C @ A )
& ( ord_le4245800335709223507ccount @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_997_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_998_just__cash__subtract,axiom,
! [A: real,B: real] :
( ( minus_4846202936726426316ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B ) )
= ( risk_Free_just_cash @ ( minus_minus_real @ A @ B ) ) ) ).
% just_cash_subtract
thf(fact_999_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1000_Icc__subset__Iic__iff,axiom,
! [L: risk_Free_account,H: risk_Free_account,H3: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or4484699493994522366ccount @ L @ H ) @ ( set_or3854930313887350124ccount @ H3 ) )
= ( ~ ( ord_le4245800335709223507ccount @ L @ H )
| ( ord_le4245800335709223507ccount @ H @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1001_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1002_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N3: nat] : ( if_real @ ( N3 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_1003_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M4: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1004_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M4: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1005_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M4: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M4 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1006_just__cash__embed,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( risk_Free_just_cash @ A3 )
= ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_embed
thf(fact_1007_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1008_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1009_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M4: nat,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ ( suc @ N2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_1010_sum__power__shift,axiom,
! [M4: nat,N2: nat,X: complex] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ X @ M4 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M4 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_1011_sum__power__shift,axiom,
! [M4: nat,N2: nat,X: real] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( times_times_real @ ( power_power_real @ X @ M4 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M4 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_1012_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1013_atLeastatMost__psubset__iff,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le5106303358561053821ccount @ ( set_or4484699493994522366ccount @ A @ B ) @ ( set_or4484699493994522366ccount @ C @ D ) )
= ( ( ~ ( ord_le4245800335709223507ccount @ A @ B )
| ( ( ord_le4245800335709223507ccount @ C @ A )
& ( ord_le4245800335709223507ccount @ B @ D )
& ( ( ord_le2131251472502387783ccount @ C @ A )
| ( ord_le2131251472502387783ccount @ B @ D ) ) ) )
& ( ord_le4245800335709223507ccount @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1014_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1015_sum__gp__multiplied,axiom,
! [M4: nat,N2: nat,X: complex] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M4 ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_1016_sum__gp__multiplied,axiom,
! [M4: nat,N2: nat,X: real] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M4 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_1017_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1018_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1019_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A3 ) @ ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_order_embed
thf(fact_1020_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_1021_sum__shift__lb__Suc0__0,axiom,
! [F: nat > risk_Free_account,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_z1425366712893667068ccount )
=> ( ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1022_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1023_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1024_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1025_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_1026_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_1027_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > complex,N2: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_1028_prod_Onat__ivl__Suc_H,axiom,
! [M4: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_1029_prod_Onat__ivl__Suc_H,axiom,
! [M4: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_1030_prod_Onat__ivl__Suc_H,axiom,
! [M4: nat,N2: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( G @ ( suc @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_1031_prod_OatLeast__Suc__atMost,axiom,
! [M4: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( times_times_real @ ( G @ M4 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_1032_prod_OatLeast__Suc__atMost,axiom,
! [M4: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( times_times_nat @ ( G @ M4 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_1033_prod_OatLeast__Suc__atMost,axiom,
! [M4: nat,N2: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( times_times_complex @ ( G @ M4 ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M4 ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_1034_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1035_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N2 )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N2 )
= A ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1036_sum__Suc__diff,axiom,
! [M4: nat,N2: nat,F: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups6033208628184776703ccount
@ ^ [I: nat] : ( minus_4846202936726426316ccount @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( minus_4846202936726426316ccount @ ( F @ ( suc @ N2 ) ) @ ( F @ M4 ) ) ) ) ).
% sum_Suc_diff
thf(fact_1037_sum__Suc__diff,axiom,
! [M4: nat,N2: nat,F: nat > complex] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( minus_minus_complex @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( minus_minus_complex @ ( F @ ( suc @ N2 ) ) @ ( F @ M4 ) ) ) ) ).
% sum_Suc_diff
thf(fact_1038_sum__Suc__diff,axiom,
! [M4: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M4 ) ) ) ) ).
% sum_Suc_diff
thf(fact_1039_prod_OSuc__reindex__ivl,axiom,
! [M4: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ M4 )
@ ( groups129246275422532515t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_1040_prod_OSuc__reindex__ivl,axiom,
! [M4: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ M4 )
@ ( groups708209901874060359at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_1041_prod_OSuc__reindex__ivl,axiom,
! [M4: nat,N2: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( G @ M4 )
@ ( groups6464643781859351333omplex
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_1042_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_1043_just__cash__valid__transfer,axiom,
! [C: real,T2: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T2 ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T2 )
& ( ord_less_eq_real @ T2 @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_1044_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C3: real] :
( risk_F5458100604530014700ccount
@ ^ [N3: nat] : ( if_real @ ( N3 = zero_zero_nat ) @ C3 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_1045_power__decreasing__iff,axiom,
! [B: real,M4: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M4 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1046_power__decreasing__iff,axiom,
! [B: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1047_power__mono__iff,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1048_power__mono__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1049_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1050_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1051_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_1052_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_1053_power__one,axiom,
! [N2: nat] :
( ( power_power_complex @ one_one_complex @ N2 )
= one_one_complex ) ).
% power_one
thf(fact_1054_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1055_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M4: nat] :
( ( ( power_power_nat @ X @ M4 )
= ( suc @ zero_zero_nat ) )
= ( ( M4 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1056_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_1057_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1058_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1059_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1060_power__inject__exp,axiom,
! [A: real,M4: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M4 )
= ( power_power_real @ A @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_1061_power__inject__exp,axiom,
! [A: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M4 )
= ( power_power_nat @ A @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_1062_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_1063_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1064_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_1065_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1066_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1067_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1068_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1069_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1070_power__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( power_power_real @ A @ N2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_1071_power__eq__0__iff,axiom,
! [A: nat,N2: nat] :
( ( ( power_power_nat @ A @ N2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_1072_power__eq__0__iff,axiom,
! [A: complex,N2: nat] :
( ( ( power_power_complex @ A @ N2 )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_1073_power__strict__decreasing__iff,axiom,
! [B: real,M4: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M4 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1074_power__strict__decreasing__iff,axiom,
! [B: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1075_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_1076_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_1077_power__not__zero,axiom,
! [A: complex,N2: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N2 )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_1078_power__commutes,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1079_power__commutes,axiom,
! [A: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1080_power__commutes,axiom,
! [A: complex,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1081_power__mult__distrib,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1082_power__mult__distrib,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1083_power__mult__distrib,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1084_power__commuting__commutes,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1085_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1086_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1087_nat__power__less__imp__less,axiom,
! [I2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M4 ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1088_power__mult,axiom,
! [A: real,M4: nat,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M4 @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ M4 ) @ N2 ) ) ).
% power_mult
thf(fact_1089_power__mult,axiom,
! [A: nat,M4: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M4 @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ M4 ) @ N2 ) ) ).
% power_mult
thf(fact_1090_power__mult,axiom,
! [A: complex,M4: nat,N2: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M4 @ N2 ) )
= ( power_power_complex @ ( power_power_complex @ A @ M4 ) @ N2 ) ) ).
% power_mult
thf(fact_1091_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_1092_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_1093_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_1094_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_1095_zero__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_1096_zero__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_1097_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_1098_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_1099_left__right__inverse__power,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_1100_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1101_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_1102_power__Suc,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_1103_power__Suc,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_1104_power__Suc,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_1105_power__Suc2,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1106_power__Suc2,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1107_power__Suc2,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1108_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_1109_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1110_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_1111_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_1112_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_1113_power__less__imp__less__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1114_power__less__imp__less__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1115_power__le__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_1116_power__le__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1117_power__le__imp__le__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1118_power__le__imp__le__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1119_power__inject__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N2 ) )
= ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1120_power__inject__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1121_power__gt1__lemma,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_1122_power__gt1__lemma,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_1123_power__less__power__Suc,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_1124_power__less__power__Suc,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_1125_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_1126_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1127_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_1128_power__gt1,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_1129_power__gt1,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_1130_power__increasing,axiom,
! [N2: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1131_power__increasing,axiom,
! [N2: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1132_power__strict__increasing,axiom,
! [N2: nat,N6: nat,A: real] :
( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1133_power__strict__increasing,axiom,
! [N2: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1134_power__less__imp__less__exp,axiom,
! [A: real,M4: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M4 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_1135_power__less__imp__less__exp,axiom,
! [A: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M4 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_1136_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_1137_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1138_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_1139_power__Suc__less,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_1140_power__Suc__less,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_1141_power__Suc__le__self,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1142_power__Suc__le__self,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1143_power__Suc__less__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_1144_power__Suc__less__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1145_power__decreasing,axiom,
! [N2: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_1146_power__decreasing,axiom,
! [N2: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_1147_power__strict__decreasing,axiom,
! [N2: nat,N6: nat,A: real] :
( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1148_power__strict__decreasing,axiom,
! [N2: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1149_power__le__imp__le__exp,axiom,
! [A: real,M4: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M4 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_1150_power__le__imp__le__exp,axiom,
! [A: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M4 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_1151_power__eq__iff__eq__base,axiom,
! [N2: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1152_power__eq__iff__eq__base,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1153_power__eq__imp__eq__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1154_power__eq__imp__eq__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1155_self__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_1156_self__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_1157_one__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_1158_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_1159_power__strict__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_1160_power__strict__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_1161_power__eq__if,axiom,
( power_power_real
= ( ^ [P6: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1162_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1163_power__eq__if,axiom,
( power_power_complex
= ( ^ [P6: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1164_power__minus__mult,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_1165_power__minus__mult,axiom,
! [N2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_1166_power__minus__mult,axiom,
! [N2: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_1167_polyfun__eq__const,axiom,
! [C: nat > complex,N2: nat,K: complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
=> ( ( C @ X3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_1168_polyfun__eq__const,axiom,
! [C: nat > real,N2: nat,K: real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
=> ( ( C @ X3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_1169_polyfun__eq__0,axiom,
! [C: nat > complex,N2: nat] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( C @ I )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_1170_polyfun__eq__0,axiom,
! [C: nat > real,N2: nat] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( C @ I )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_1171_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N2: nat,K: nat] :
( ! [W: complex] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ W @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( C @ K )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_1172_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N2: nat,K: nat] :
( ! [W: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ W @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_1173_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N2: nat,D: nat > complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( D @ I ) @ ( power_power_complex @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( C @ I )
= ( D @ I ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_1174_polyfun__eq__coeffs,axiom,
! [C: nat > real,N2: nat,D: nat > real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( D @ I ) @ ( power_power_real @ X3 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( C @ I )
= ( D @ I ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_1175_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_1176_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_1177_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_1178_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( P @ M5 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1179_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( P @ M5 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1180_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > complex,N2: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z4: complex] :
( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z4 ) )
=> ( ord_less_eq_real
@ ( real_V1022390504157884413omplex
@ ( groups2073611262835488442omplex
@ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ Z4 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_1181_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z4: real] :
( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z4 ) )
=> ( ord_less_eq_real
@ ( real_V7735802525324610683m_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ Z4 @ I ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_1182_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_1183_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_1184_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_1185_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_1186_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_1187_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_1188_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_1189_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_1190_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_1191_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1192_Rep__account__uminus,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( ^ [N3: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N3 ) ) ) ) ).
% Rep_account_uminus
thf(fact_1193_just__cash__uminus,axiom,
! [A: real] :
( ( uminus3377898441596595772ccount @ ( risk_Free_just_cash @ A ) )
= ( risk_Free_just_cash @ ( uminus_uminus_real @ A ) ) ) ).
% just_cash_uminus
thf(fact_1194_return__loans__uminus,axiom,
! [Rho: nat > real,Alpha: risk_Free_account] :
( ( risk_F2121631595377017831_loans @ Rho @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( uminus3377898441596595772ccount @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) ) ) ).
% return_loans_uminus
thf(fact_1195_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N3: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N3 ) ) ) ) ) ).
% uminus_account_def
thf(fact_1196_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_1197_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1198_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_1199_sum__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1200_sumr__cos__zero__one,axiom,
! [N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_1201_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_1202_add__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1203_add__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ M4 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1204_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1205_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1206_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1207_add__gr__0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1208_mult__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ M4 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1209_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1210_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1211_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1212_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_1213_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1214_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1215_one__is__add,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1216_add__is__1,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1217_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1218_less__natE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M4 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1219_less__add__Suc1,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M4 ) ) ) ).
% less_add_Suc1
thf(fact_1220_less__add__Suc2,axiom,
! [I2: nat,M4: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M4 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1221_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1222_less__imp__Suc__add,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1223_mono__nat__linear__lb,axiom,
! [F: nat > nat,M4: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M4 ) @ K ) @ ( F @ ( plus_plus_nat @ M4 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1224_diff__add__0,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1225_mult__Suc,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1226_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1227_add__diff__inverse__nat,axiom,
! [M4: nat,N2: nat] :
( ~ ( ord_less_nat @ M4 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M4 @ N2 ) )
= M4 ) ) ).
% add_diff_inverse_nat
thf(fact_1228_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1229_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1230_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1231_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1232_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1233_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1234_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1235_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1236_left__add__mult__distrib,axiom,
! [I2: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1237_add__mult__distrib2,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M4 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1238_add__mult__distrib,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1239_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1240_diff__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_cancel2
thf(fact_1241_diff__add__inverse,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M4 ) @ N2 )
= M4 ) ).
% diff_add_inverse
thf(fact_1242_diff__add__inverse2,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N2 ) @ N2 )
= M4 ) ).
% diff_add_inverse2
thf(fact_1243_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M5 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1244_trans__le__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1245_trans__le__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1246_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1247_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1248_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1249_add__leD2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1250_add__leD1,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% add_leD1
thf(fact_1251_le__add2,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M4 @ N2 ) ) ).
% le_add2
thf(fact_1252_le__add1,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) ) ).
% le_add1
thf(fact_1253_add__leE,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M4 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1254_less__add__eq__less,axiom,
! [K: nat,L: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1255_trans__less__add2,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_1256_trans__less__add1,axiom,
! [I2: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_1257_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1258_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1259_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1260_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1261_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1262_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1263_add__Suc,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc
thf(fact_1264_add__Suc__shift,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ M4 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( rho @ I ) ) @ ( risk_F170160801229183585ccount @ alpha2 @ I ) )
@ ( set_ord_atMost_nat @ n ) ) ) ).
%------------------------------------------------------------------------------