TPTP Problem File: SLH0876^1.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_02043_064947__6070180_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1366 ( 615 unt; 92 typ; 0 def)
% Number of atoms : 3575 (1561 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11654 ( 236 ~; 80 |; 251 &;9789 @)
% ( 0 <=>;1298 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 367 ( 367 >; 0 *; 0 +; 0 <<)
% Number of symbols : 86 ( 83 usr; 14 con; 0-4 aty)
% Number of variables : 3413 ( 138 ^;3241 !; 34 ?;3413 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:58:23.145
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_Ri1641125681238393385ccount: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__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 (83)
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $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_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__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Risk____Free____Lending__Oaccount,type,
plus_p1863581527469039996ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
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_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__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_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_Nat_OSuc,type,
suc: nat > nat ).
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_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
ord_le4487465848215339657ccount: set_Ri1641125681238393385ccount > set_Ri1641125681238393385ccount > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $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_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_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_Obulk__update__account,type,
risk_F2412532053715321062ccount: nat > ( nat > real ) > real > risk_Free_account > risk_Free_account ).
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_Oloan,type,
risk_Free_loan: nat > real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Onet__asset__value,type,
risk_F2906766666041932210_value: risk_Free_account > real ).
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_Oshortest__period,type,
risk_F4612863212915232279period: risk_Free_account > nat ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Risk__Free__Lending_Oupdate__account,type,
risk_F444380041991734328ccount: ( nat > real ) > real > risk_Free_account > risk_Free_account ).
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__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_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
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_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v__092_060alpha_062,type,
alpha: risk_Free_account ).
thf(sy_v__092_060nu_062_H____,type,
nu: nat > real ).
thf(sy_v__092_060nu_062____,type,
nu2: nat > real ).
thf(sy_v__092_060rho_062,type,
rho: nat > real ).
thf(sy_v_i,type,
i: real ).
thf(sy_v_ka____,type,
ka: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1266)
thf(fact_0__092_060open_0620_A_060_Ak_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ka ).
% \<open>0 < k\<close>
thf(fact_1__092_060open_062k_A_092_060le_062_Am_092_060close_062,axiom,
ord_less_eq_nat @ ka @ m ).
% \<open>k \<le> m\<close>
thf(fact_2_assms_I2_J,axiom,
! [N: nat] : ( ord_less_real @ ( rho @ N ) @ one_one_real ) ).
% assms(2)
thf(fact_3__092_060open_062k_A_092_060le_062_Ashortest__period_A_092_060alpha_062_092_060close_062,axiom,
ord_less_eq_nat @ ka @ ( risk_F4612863212915232279period @ alpha ) ).
% \<open>k \<le> shortest_period \<alpha>\<close>
thf(fact_4__092_060open_062_092_060nu_062_Ak_A_061_A_I1_A_N_A_092_060rho_062_Ak_J_A_094_An_A_K_A_092_060pi_062_A_092_060alpha_062_Ak_092_060close_062,axiom,
( ( nu2 @ ka )
= ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ ka ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ ka ) ) ) ).
% \<open>\<nu> k = (1 - \<rho> k) ^ n * \<pi> \<alpha> k\<close>
thf(fact_5_assms_I4_J,axiom,
( ( rho @ zero_zero_nat )
= zero_zero_real ) ).
% assms(4)
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_calculation,axiom,
( ( risk_F170160801229183585ccount
@ ( groups6033208628184776703ccount
@ ^ [J: nat] : ( risk_Free_loan @ J @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ J ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ m ) )
@ ka )
= ( groups6591440286371151544t_real
@ ^ [J: nat] : ( risk_F170160801229183585ccount @ ( risk_Free_loan @ J @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ J ) ) ) @ ka )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ m ) ) ) ).
% calculation
thf(fact_8_assms_I3_J,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ord_less_real @ ( rho @ N ) @ ( rho @ M ) ) ) ).
% assms(3)
thf(fact_9_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_10_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_11_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_12_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_13_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_14_power__one,axiom,
! [N2: nat] :
( ( power_power_complex @ one_one_complex @ N2 )
= one_one_complex ) ).
% power_one
thf(fact_15_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_16_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_17_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_18_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_19_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_20_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_21_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_22_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_23_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_24_return__loans__loan,axiom,
! [Rho: nat > real,N2: nat,C: real] :
( ( risk_F2121631595377017831_loans @ Rho @ ( risk_Free_loan @ N2 @ C ) )
= ( risk_Free_loan @ N2 @ ( times_times_real @ ( minus_minus_real @ one_one_real @ ( Rho @ N2 ) ) @ C ) ) ) ).
% return_loans_loan
thf(fact_25_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_26_sum__subtractf,axiom,
! [F: nat > risk_Free_account,G: nat > risk_Free_account,A2: set_nat] :
( ( groups6033208628184776703ccount
@ ^ [X2: nat] : ( minus_4846202936726426316ccount @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_4846202936726426316ccount @ ( groups6033208628184776703ccount @ F @ A2 ) @ ( groups6033208628184776703ccount @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_27_sum__subtractf,axiom,
! [F: complex > complex,G: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_28_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_29_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_30_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_31_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_32_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_33_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_34_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_35_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_36_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_37_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_38_diff__zero,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_zero
thf(fact_39_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_40_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_41_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_42_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_43_diff__0__right,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_0_right
thf(fact_44_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_45_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_46_diff__self,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ A )
= zero_z1425366712893667068ccount ) ).
% diff_self
thf(fact_47_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_48_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_49_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6033208628184776703ccount
@ ^ [Uu: nat] : zero_z1425366712893667068ccount
@ A2 )
= zero_z1425366712893667068ccount ) ).
% sum.neutral_const
thf(fact_50_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_51_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_52_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_53_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_54_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_55_power__inject__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_56_power__inject__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_57_Rep__account__loan,axiom,
! [N2: nat,X: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_loan @ N2 @ X ) )
= ( ^ [M3: nat] : ( if_real @ ( N2 = M3 ) @ X @ zero_zero_real ) ) ) ).
% Rep_account_loan
thf(fact_58_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_59_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_60_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_61_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_62_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_63_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_64_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_65_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_66_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_67_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_68_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_69__092_060open_062_092_060forall_062k_062shortest__period_A_092_060alpha_062_O_A_092_060nu_062_Ak_A_061_A_092_060nu_062_H_Ak_092_060close_062,axiom,
! [K: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ alpha ) @ K )
=> ( ( nu2 @ K )
= ( nu @ K ) ) ) ).
% \<open>\<forall>k>shortest_period \<alpha>. \<nu> k = \<nu>' k\<close>
thf(fact_70_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_71_power__decreasing__iff,axiom,
! [B: nat,M2: 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 @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_72_power__decreasing__iff,axiom,
! [B: real,M2: 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 @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_73_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_74_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_75_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_76_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_83_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_84_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_85_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_86_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_87_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_88_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_89_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_90_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_91_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_92_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_93_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_94_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_95_power__mult,axiom,
! [A: real,M2: nat,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_96_power__mult,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_97_power__mult,axiom,
! [A: complex,M2: nat,N2: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_complex @ ( power_power_complex @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_98_greater__than__shortest__period__zero,axiom,
! [Alpha: risk_Free_account,M2: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ Alpha ) @ M2 )
=> ( ( risk_F170160801229183585ccount @ Alpha @ M2 )
= zero_zero_real ) ) ).
% greater_than_shortest_period_zero
thf(fact_99_shortest__period__bound,axiom,
! [Alpha: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real )
=> ( ord_less_eq_nat @ I @ ( risk_F4612863212915232279period @ Alpha ) ) ) ).
% shortest_period_bound
thf(fact_100_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_101_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_102_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_103_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_104_shortest__period___092_060pi_062,axiom,
! [Alpha: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real )
=> ( ( risk_F170160801229183585ccount @ Alpha @ ( risk_F4612863212915232279period @ Alpha ) )
!= zero_zero_real ) ) ).
% shortest_period_\<pi>
thf(fact_105_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_106_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_107_shortest__period__loan,axiom,
! [C: real,N2: nat] :
( ( C != zero_zero_real )
=> ( ( risk_F4612863212915232279period @ ( risk_Free_loan @ N2 @ C ) )
= N2 ) ) ).
% shortest_period_loan
thf(fact_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_122_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_123_sum__nonpos,axiom,
! [A2: set_real,F: real > risk_Free_account] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ A2 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_124_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_125_sum__nonpos,axiom,
! [A2: set_nat,F: nat > risk_Free_account] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ A2 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_126_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_127_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_128_sum__nonneg,axiom,
! [A2: set_real,F: real > risk_Free_account] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X3 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups8516999891779824987ccount @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_129_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_130_sum__nonneg,axiom,
! [A2: set_nat,F: nat > risk_Free_account] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X3 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups6033208628184776703ccount @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: complex,Z: complex] : ( Y2 = Z ) )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_151_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_152_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ B2 )
= zero_z1425366712893667068ccount ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_153_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_154_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_155_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_156_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_157_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_158_power__strict__decreasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_159_power__strict__decreasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_160_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_161_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_162_power__decreasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_163_power__decreasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_164_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_165_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_166_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_167_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_168_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_169_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > risk_Free_account,A2: set_nat] :
( ( ( groups6033208628184776703ccount @ G @ A2 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_170_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_171_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_172_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > risk_Free_account] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups6033208628184776703ccount @ G @ A2 )
= zero_z1425366712893667068ccount ) ) ).
% sum.neutral
thf(fact_173_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_174_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_175_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_176_power__strict__increasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_177_power__strict__increasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_178_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_179_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_180_power__increasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_181_power__increasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_182_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_183_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_184_power__eq__if,axiom,
( power_power_complex
= ( ^ [P2: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P2 @ ( power_power_complex @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_185_power__eq__if,axiom,
( power_power_real
= ( ^ [P2: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P2 @ ( power_power_real @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_186_power__eq__if,axiom,
( power_power_nat
= ( ^ [P2: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_187_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_188_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_189_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_190_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_191_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_192_sum__mono,axiom,
! [K2: set_real,F: real > risk_Free_account,G: real > risk_Free_account] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ K2 ) @ ( groups8516999891779824987ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_193_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_194_sum__mono,axiom,
! [K2: set_nat,F: nat > risk_Free_account,G: nat > risk_Free_account] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ K2 ) @ ( groups6033208628184776703ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_195_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_196_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_197_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_198_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_199_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_200_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_201_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_202_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_203_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_204_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_205_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_206_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_207_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_208_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_209_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_210_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_211_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J2: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_212_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J2: real > nat,T: set_nat,H: nat > risk_Free_account,G: real > risk_Free_account] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups6033208628184776703ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_213_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: complex > real,J2: real > complex,T: set_complex,H: complex > complex,G: real > complex] :
( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( member_complex @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_214_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J2: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_real @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_215_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J2: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_216_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J2: nat > real,T: set_real,H: real > risk_Free_account,G: nat > risk_Free_account] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_real @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups8516999891779824987ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_217_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J2: nat > nat,T: set_nat,H: nat > risk_Free_account,G: nat > risk_Free_account] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups6033208628184776703ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_218_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I: real > complex,J2: complex > real,T: set_real,H: real > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( member_real @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_complex @ ( I @ B3 ) @ S ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups5754745047067104278omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_219_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I: complex > complex,J2: complex > complex,T: set_complex,H: complex > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( I @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( member_complex @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_complex @ ( I @ B3 ) @ S ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_220_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K3: nat > real,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( ( member_real @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_221_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K3: nat > real,A2: set_real,H: real > nat,Gamma: nat > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( ( member_real @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A2 )
= ( groups6033208628184776703ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_222_sum_Oeq__general__inverses,axiom,
! [B4: set_complex,K3: complex > real,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B4 )
=> ( ( member_real @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_223_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K3: real > nat,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_224_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K3: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_225_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K3: real > nat,A2: set_nat,H: nat > real,Gamma: real > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A2 )
= ( groups8516999891779824987ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_226_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K3: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( ( member_nat @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A2 )
= ( groups6033208628184776703ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_227_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K3: real > complex,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ( ( member_complex @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_228_sum_Oeq__general__inverses,axiom,
! [B4: set_complex,K3: complex > complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B4 )
=> ( ( member_complex @ ( K3 @ Y3 ) @ A2 )
& ( ( H @ ( K3 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B4 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_229_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_230_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_real,H: real > nat,Gamma: nat > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A2 )
= ( groups6033208628184776703ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_231_sum_Oeq__general,axiom,
! [B4: set_complex,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_232_sum_Oeq__general,axiom,
! [B4: set_real,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_233_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_234_sum_Oeq__general,axiom,
! [B4: set_real,A2: set_nat,H: nat > real,Gamma: real > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A2 )
= ( groups8516999891779824987ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_235_sum_Oeq__general,axiom,
! [B4: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A2 )
= ( groups6033208628184776703ccount @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_236_sum_Oeq__general,axiom,
! [B4: set_real,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B4 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_237_sum_Oeq__general,axiom,
! [B4: set_complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B4 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_238_sum_Ocong,axiom,
! [A2: set_nat,B4: set_nat,G: nat > real,H: nat > real] :
( ( A2 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_239_sum_Ocong,axiom,
! [A2: set_nat,B4: set_nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( A2 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ A2 )
= ( groups6033208628184776703ccount @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_240_sum_Ocong,axiom,
! [A2: set_complex,B4: set_complex,G: complex > complex,H: complex > complex] :
( ( A2 = B4 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= ( groups7754918857620584856omplex @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_241_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_242_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_243_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_244_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_245_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_246_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_247_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_248_sum_Oswap,axiom,
! [G: nat > nat > real,B4: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B4 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ J )
@ A2 )
@ B4 ) ) ).
% sum.swap
thf(fact_249_sum_Oswap,axiom,
! [G: nat > nat > risk_Free_account,B4: set_nat,A2: set_nat] :
( ( groups6033208628184776703ccount
@ ^ [I3: nat] : ( groups6033208628184776703ccount @ ( G @ I3 ) @ B4 )
@ A2 )
= ( groups6033208628184776703ccount
@ ^ [J: nat] :
( groups6033208628184776703ccount
@ ^ [I3: nat] : ( G @ I3 @ J )
@ A2 )
@ B4 ) ) ).
% sum.swap
thf(fact_250_sum_Oswap,axiom,
! [G: complex > complex > complex,B4: set_complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B4 )
@ A2 )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I3: complex] : ( G @ I3 @ J )
@ A2 )
@ B4 ) ) ).
% sum.swap
thf(fact_251_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_252_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_253_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_254_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_255_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_256_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_257_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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
@ ^ [I3: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_271_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
@ ^ [I3: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J ) )
@ B4 )
@ A2 ) ) ).
% sum_product
thf(fact_272_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_273_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_274_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_275_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_276_mult__le__cancel2,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_277_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_278_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_279_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_289_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N2 )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N2 )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_290_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_291_assms_I1_J,axiom,
ord_less_eq_real @ zero_zero_real @ i ).
% assms(1)
thf(fact_292_return__loans__zero,axiom,
! [Rho: nat > real] :
( ( risk_F2121631595377017831_loans @ Rho @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% return_loans_zero
thf(fact_293_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_294_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_295_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_296_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_297_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_298_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_309_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_310_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_311_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_312_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_313_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_314_mult__cancel2,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K3 )
= ( times_times_nat @ N2 @ K3 ) )
= ( ( M2 = N2 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_315_mult__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N2 ) )
= ( ( M2 = N2 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_316_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_317_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_318_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_319_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_320_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_321_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_322_loan__zero,axiom,
! [N2: nat] :
( ( risk_Free_loan @ N2 @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% loan_zero
thf(fact_323_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_324_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_325_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_326_mult__less__cancel2,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_327_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_328_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_329_diff__commute,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J2 ) ) ).
% diff_commute
thf(fact_330_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_331_diff__mult__distrib2,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_332_return__loans__mono,axiom,
! [Rho: nat > real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ord_less_eq_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( ord_le4245800335709223507ccount @ Alpha @ Beta )
=> ( ord_le4245800335709223507ccount @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) @ ( risk_F2121631595377017831_loans @ Rho @ Beta ) ) ) ) ) ).
% return_loans_mono
thf(fact_333_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_334_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N2 ) )
= ( ( K3 = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_335_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_336_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_337_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_338_less__imp__diff__less,axiom,
! [J2: nat,K3: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_339_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_340_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N5 )
& ~ ( P @ M ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_341_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N5 )
=> ( P @ M ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_342_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_343_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_344_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_345_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_346_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_347_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_348_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_349_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_350_mult__le__mono2,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_351_mult__le__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_352_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_353_mult__le__mono,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_354_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_355_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_356_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_357_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N2 @ K3 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_358_le__diff__iff,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N2 @ K3 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_359_eq__diff__iff,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K3 )
= ( minus_minus_nat @ N2 @ K3 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_360_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_361_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_362_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_363_le__trans,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_364_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_365_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_366_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_367_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_368_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_369_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_370_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_371_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_388_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_389_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_390_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_391_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_392_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_393_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_394_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_395_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_396_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_397_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_398_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_399_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_400_nat__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_401_nat__mult__eq__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_402_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N5 )
& ~ ( P @ M ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_403_mult__less__mono2,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_404_mult__less__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_405_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_406_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_407_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_408_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_409_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_410_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_411_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_412_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_413_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_414_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_415_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_416_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_417_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_418_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_419_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_420_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_421_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_422_less__diff__iff,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N2 @ K3 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_423_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_424_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_425_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_426_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_427_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_428_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_429_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_430_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_431_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_432_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_433_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_434_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_435_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_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_462_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_463_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_464_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_465_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_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_492_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_493_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_494_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_495_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_496_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_497_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_498_less__1__mult,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_499_less__1__mult,axiom,
! [M2: real,N2: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_500_nat__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_501_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H3: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_541_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_542_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_543_atLeastAtMost__iff,axiom,
! [I: risk_Free_account,L: risk_Free_account,U: risk_Free_account] :
( ( member5612106785598075018ccount @ I @ ( set_or4484699493994522366ccount @ L @ U ) )
= ( ( ord_le4245800335709223507ccount @ L @ I )
& ( ord_le4245800335709223507ccount @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_544_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_545_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_546_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_547_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_548_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_549_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_550_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_551__092_060open_062_092_060forall_062c_O_A_092_060pi_062_A_Ijust__cash_Ac_J_Ak_A_061_A0_092_060close_062,axiom,
! [C2: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C2 ) @ ka )
= zero_zero_real ) ).
% \<open>\<forall>c. \<pi> (just_cash c) k = 0\<close>
thf(fact_552_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N5: nat] : ( ord_less_real @ ( power_power_real @ X @ N5 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_553_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_554_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_555_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_556_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_557_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_558_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_559_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_560_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_561_just__cash__embed,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( risk_Free_just_cash @ A3 )
= ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_embed
thf(fact_562_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_563_loan__just__cash,axiom,
! [C: real] :
( ( risk_Free_loan @ zero_zero_nat @ C )
= ( risk_Free_just_cash @ C ) ) ).
% loan_just_cash
thf(fact_564_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% linordered_field_no_lb
thf(fact_565_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_566_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_567_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_568_return__loans__just__cash,axiom,
! [Rho: nat > real,C: real] :
( ( ( Rho @ zero_zero_nat )
= zero_zero_real )
=> ( ( risk_F2121631595377017831_loans @ Rho @ ( risk_Free_just_cash @ C ) )
= ( risk_Free_just_cash @ C ) ) ) ).
% return_loans_just_cash
thf(fact_569_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_570_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_571_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_572_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_573_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_574_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_575_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N5: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N5 ) ) ) ).
% real_arch_pow
thf(fact_576_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_577_X,axiom,
( nu2
= ( ^ [K5: nat] :
( if_real @ ( K5 = zero_zero_nat )
@ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ ( risk_F1914734008469130493eserve @ alpha ) )
@ ( groups6591440286371151544t_real
@ ^ [J: nat] : ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( risk_F170160801229183585ccount @ alpha @ J ) @ i ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) ) ) @ ( plus_plus_real @ i @ ( rho @ J ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ ( risk_F4612863212915232279period @ alpha ) ) ) )
@ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ K5 ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ K5 ) ) ) ) ) ).
% X
thf(fact_578_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B3: real,C3: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A4 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_579_Y,axiom,
( nu
= ( risk_F170160801229183585ccount
@ ( plus_p1863581527469039996ccount
@ ( risk_Free_just_cash
@ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ ( risk_F1914734008469130493eserve @ alpha ) )
@ ( groups6591440286371151544t_real
@ ^ [J: nat] : ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( risk_F170160801229183585ccount @ alpha @ J ) @ i ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) ) ) @ ( plus_plus_real @ i @ ( rho @ J ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ ( risk_F4612863212915232279period @ alpha ) ) ) ) )
@ ( groups6033208628184776703ccount
@ ^ [J: nat] : ( risk_Free_loan @ J @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ J ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ ( risk_F4612863212915232279period @ alpha ) ) ) ) ) ) ).
% Y
thf(fact_580_update__account__mono,axiom,
! [I: real,Rho: nat > real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ord_less_eq_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( ord_le4245800335709223507ccount @ Alpha @ Beta )
=> ( ord_le4245800335709223507ccount @ ( risk_F444380041991734328ccount @ Rho @ I @ Alpha ) @ ( risk_F444380041991734328ccount @ Rho @ I @ Beta ) ) ) ) ) ) ).
% update_account_mono
thf(fact_581_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_582_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_583_add__left__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_584_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_585_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_586_add__right__cancel,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_587_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_588_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_589_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_590_add__le__cancel__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_591_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_592_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_593_add__le__cancel__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
= ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_594_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_595_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_596_add__0,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add_0
thf(fact_597_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_598_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_599_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_600_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_601_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_602_add__cancel__right__right,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ A @ B ) )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_right
thf(fact_603_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_604_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_605_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_606_add__cancel__right__left,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ B @ A ) )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_left
thf(fact_607_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_608_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_609_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_610_add__cancel__left__right,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= A )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_right
thf(fact_611_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_612_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_613_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_614_add__cancel__left__left,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= A )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_left
thf(fact_615_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_616_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_617_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_618_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_619_add_Oright__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.right_neutral
thf(fact_620_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_621_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_622_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_623_add__less__cancel__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_624_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_625_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_626_add__less__cancel__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
= ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_627_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_628_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_629_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_630_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_631_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_632_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_633_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_634_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_635_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_636_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_637_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_638_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_639_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_640_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_641_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_642_add__diff__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_643_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_644_diff__add__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_645_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_646_add__diff__cancel__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
= ( minus_4846202936726426316ccount @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_647_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_648_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_649_add__diff__cancel__left_H,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_650_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_651_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_652_add__diff__cancel__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( minus_4846202936726426316ccount @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_653_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_654_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_655_add__diff__cancel__right_H,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_656_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_657_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_658_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_659_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_660_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_661_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_662_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_663_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_664_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_665_update__account__zero,axiom,
! [Rho: nat > real,I: real] :
( ( risk_F444380041991734328ccount @ Rho @ I @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% update_account_zero
thf(fact_666_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_667_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_668_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_669_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_670_le__add__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( plus_p1863581527469039996ccount @ B @ A ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% le_add_same_cancel2
thf(fact_671_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_672_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_673_le__add__same__cancel1,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( plus_p1863581527469039996ccount @ A @ B ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% le_add_same_cancel1
thf(fact_674_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_675_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_676_add__le__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel2
thf(fact_677_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_678_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_679_add__le__same__cancel1,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ B @ A ) @ B )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel1
thf(fact_680_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_681_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_682_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_683_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_684_less__add__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( plus_p1863581527469039996ccount @ B @ A ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% less_add_same_cancel2
thf(fact_685_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_686_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_687_less__add__same__cancel1,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( plus_p1863581527469039996ccount @ A @ B ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% less_add_same_cancel1
thf(fact_688_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_689_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_690_add__less__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel2
thf(fact_691_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_692_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_693_add__less__same__cancel1,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ B @ A ) @ B )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel1
thf(fact_694_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_695_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_696_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_697_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_698_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_699_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_700_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_701_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_702_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_703_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_704_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_705_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_706_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_707_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_708_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_709_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_710_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_711_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_712_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_713_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_714_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_715_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_716_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_717_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_718_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_719_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_720_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_721_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_722_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_723_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_724_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_725_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_726_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_727_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_728_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_729_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_730_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_731_Rep__account__plus,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) )
= ( ^ [N3: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N3 ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N3 ) ) ) ) ).
% Rep_account_plus
thf(fact_732_just__cash__plus,axiom,
! [A: real,B: real] :
( ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B ) )
= ( risk_Free_just_cash @ ( plus_plus_real @ A @ B ) ) ) ).
% just_cash_plus
thf(fact_733_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_734_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_735_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_736_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_737_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_738_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_739_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_740_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_741_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_742_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_743_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_744_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_745_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_746_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_747_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_748_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_749_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_750_add__divide__eq__if__simps_I2_J,axiom,
! [Z3: complex,A: complex,B: complex] :
( ( ( Z3 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
= B ) )
& ( ( Z3 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_751_add__divide__eq__if__simps_I2_J,axiom,
! [Z3: real,A: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
= B ) )
& ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_752_add__divide__eq__if__simps_I1_J,axiom,
! [Z3: complex,A: complex,B: complex] :
( ( ( Z3 = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
= A ) )
& ( ( Z3 != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_753_add__divide__eq__if__simps_I1_J,axiom,
! [Z3: real,A: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= A ) )
& ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_754_add__frac__eq,axiom,
! [Y: complex,Z3: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z3 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z3 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).
% add_frac_eq
thf(fact_755_add__frac__eq,axiom,
! [Y: real,Z3: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).
% add_frac_eq
thf(fact_756_add__frac__num,axiom,
! [Y: complex,X: complex,Z3: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z3 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_757_add__frac__num,axiom,
! [Y: real,X: real,Z3: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z3 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_758_add__num__frac,axiom,
! [Y: complex,Z3: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_759_add__num__frac,axiom,
! [Y: real,Z3: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z3 @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_760_add__divide__eq__iff,axiom,
! [Z3: complex,X: complex,Y: complex] :
( ( Z3 != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).
% add_divide_eq_iff
thf(fact_761_add__divide__eq__iff,axiom,
! [Z3: real,X: real,Y: real] :
( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).
% add_divide_eq_iff
thf(fact_762_divide__add__eq__iff,axiom,
! [Z3: complex,X: complex,Y: complex] :
( ( Z3 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).
% divide_add_eq_iff
thf(fact_763_divide__add__eq__iff,axiom,
! [Z3: real,X: real,Y: real] :
( ( Z3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).
% divide_add_eq_iff
thf(fact_764_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_765_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_766_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_767_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_768_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_769_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( I = J2 )
& ( K3 = L ) )
=> ( ( plus_plus_real @ I @ K3 )
= ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_770_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( I = J2 )
& ( K3 = L ) )
=> ( ( plus_p1863581527469039996ccount @ I @ K3 )
= ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_771_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( I = J2 )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I @ K3 )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_772_group__cancel_Oadd1,axiom,
! [A2: real,K3: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K3 @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K3 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_773_group__cancel_Oadd1,axiom,
! [A2: risk_Free_account,K3: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( A2
= ( plus_p1863581527469039996ccount @ K3 @ A ) )
=> ( ( plus_p1863581527469039996ccount @ A2 @ B )
= ( plus_p1863581527469039996ccount @ K3 @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_774_group__cancel_Oadd1,axiom,
! [A2: nat,K3: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_775_group__cancel_Oadd2,axiom,
! [B4: real,K3: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K3 @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K3 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_776_group__cancel_Oadd2,axiom,
! [B4: risk_Free_account,K3: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( B4
= ( plus_p1863581527469039996ccount @ K3 @ B ) )
=> ( ( plus_p1863581527469039996ccount @ A @ B4 )
= ( plus_p1863581527469039996ccount @ K3 @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_777_group__cancel_Oadd2,axiom,
! [B4: nat,K3: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_778_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_779_add_Oassoc,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add.assoc
thf(fact_780_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_781_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_782_add_Oleft__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_783_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_784_add_Oright__cancel,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_785_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_786_add_Ocommute,axiom,
( plus_p1863581527469039996ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( plus_p1863581527469039996ccount @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_787_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_788_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_789_add_Oleft__commute,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add.left_commute
thf(fact_790_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_791_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_792_add__left__imp__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_793_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_794_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_795_add__right__imp__eq,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_796_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_797_update__account__plus,axiom,
! [Rho: nat > real,I: real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F444380041991734328ccount @ Rho @ I @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) )
= ( plus_p1863581527469039996ccount @ ( risk_F444380041991734328ccount @ Rho @ I @ Alpha ) @ ( risk_F444380041991734328ccount @ Rho @ I @ Beta ) ) ) ).
% update_account_plus
thf(fact_798_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N3: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N3 ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N3 ) ) ) ) ) ).
% plus_account_def
thf(fact_799_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_800_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_801_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z3: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z3 ) ) ) ).
% divide_divide_times_eq
thf(fact_802_times__divide__times__eq,axiom,
! [X: real,Y: real,Z3: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_803_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_804_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_805_add__le__imp__le__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_806_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_807_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_808_add__le__imp__le__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_809_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_810_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_811_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_812_add__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add_right_mono
thf(fact_813_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_814_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_815_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_816_add__left__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% add_left_mono
thf(fact_817_add__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 @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_818_add__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 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_819_add__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ C @ D )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_mono
thf(fact_820_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_821_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I @ J2 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_822_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J2 )
& ( ord_le4245800335709223507ccount @ K3 @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_823_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_824_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( I = J2 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_825_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( I = J2 )
& ( ord_le4245800335709223507ccount @ K3 @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_826_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_827_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I @ J2 )
& ( K3 = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_828_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J2 )
& ( K3 = L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_829_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_830_add_Ogroup__left__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add.group_left_neutral
thf(fact_831_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_832_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_833_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_834_add_Ocomm__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.comm_neutral
thf(fact_835_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_836_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_837_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_838_comm__monoid__add__class_Oadd__0,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_839_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_840_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_841_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_842_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_843_add__less__imp__less__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_844_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_845_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_846_add__less__imp__less__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_847_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_848_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_849_add__strict__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_850_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_851_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_852_add__strict__left__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_853_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_854_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_855_add__strict__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_856_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K3 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_857_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_real @ I @ J2 )
& ( K3 = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_858_add__mono__thms__linordered__field_I1_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J2 )
& ( K3 = L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_859_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_860_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( I = J2 )
& ( ord_less_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_861_add__mono__thms__linordered__field_I2_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( I = J2 )
& ( ord_le2131251472502387783ccount @ K3 @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_862_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_863_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_real @ I @ J2 )
& ( ord_less_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_864_add__mono__thms__linordered__field_I5_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J2 )
& ( ord_le2131251472502387783ccount @ K3 @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_865_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_866_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_867_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_868_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_869_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_870_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_871_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_872_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_873_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_874_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_875_power__divide,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_divide
thf(fact_876_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_877_group__cancel_Osub1,axiom,
! [A2: real,K3: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K3 @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K3 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_878_group__cancel_Osub1,axiom,
! [A2: risk_Free_account,K3: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( A2
= ( plus_p1863581527469039996ccount @ K3 @ A ) )
=> ( ( minus_4846202936726426316ccount @ A2 @ B )
= ( plus_p1863581527469039996ccount @ K3 @ ( minus_4846202936726426316ccount @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_879_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_880_diff__eq__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B )
= C )
= ( A
= ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_881_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_882_eq__diff__eq,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( A
= ( minus_4846202936726426316ccount @ C @ B ) )
= ( ( plus_p1863581527469039996ccount @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_883_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_884_add__diff__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ ( minus_4846202936726426316ccount @ B @ C ) )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_885_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_886_diff__diff__eq2,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ ( minus_4846202936726426316ccount @ B @ C ) )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_887_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_888_diff__add__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ C )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_889_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_890_diff__add__eq__diff__diff__swap,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_891_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_892_add__implies__diff,axiom,
! [C: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ C @ B )
= A )
=> ( C
= ( minus_4846202936726426316ccount @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_893_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_894_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_895_diff__diff__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ C )
= ( minus_4846202936726426316ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_896_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_897_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_898_add__diff__add,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account,D: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) )
= ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ ( minus_4846202936726426316ccount @ C @ D ) ) ) ).
% add_diff_add
thf(fact_899_return__loans__plus,axiom,
! [Rho: nat > real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2121631595377017831_loans @ Rho @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) )
= ( plus_p1863581527469039996ccount @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) @ ( risk_F2121631595377017831_loans @ Rho @ Beta ) ) ) ).
% return_loans_plus
thf(fact_900_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_901_sum__divide__distrib,axiom,
! [F: complex > complex,A2: set_complex,R: complex] :
( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_902_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_903_sum_Odistrib,axiom,
! [G: nat > risk_Free_account,H: nat > risk_Free_account,A2: set_nat] :
( ( groups6033208628184776703ccount
@ ^ [X2: nat] : ( plus_p1863581527469039996ccount @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ A2 ) @ ( groups6033208628184776703ccount @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_904_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_905_scaling__mono,axiom,
! [U: real,V: real,R: real,S2: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_906_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_907_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_908_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_909_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_910_divide__nonneg__nonneg,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 @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_911_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_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_divide_iff
thf(fact_912_divide__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 @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_913_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_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 ) ) ) ) ).
% divide_le_0_iff
thf(fact_914_divide__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 @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_915_divide__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 @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_916_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_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_divide_iff
thf(fact_917_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_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 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_918_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_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 ) ) ) ) ).
% divide_less_0_iff
thf(fact_919_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_920_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_921_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_922_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_923_frac__eq__eq,axiom,
! [Y: complex,Z3: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z3 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W @ Z3 ) )
= ( ( times_times_complex @ X @ Z3 )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_924_frac__eq__eq,axiom,
! [Y: real,Z3: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z3 ) )
= ( ( times_times_real @ X @ Z3 )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_925_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_926_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_927_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_928_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_929_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_930_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_931_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_932_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_933_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_934_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_935_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_936_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_937_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_938_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_939_sum_Oub__add__nat,axiom,
! [M2: nat,N2: nat,G: nat > nat,P3: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_940_sum_Oub__add__nat,axiom,
! [M2: nat,N2: nat,G: nat > real,P3: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_941_sum_Oub__add__nat,axiom,
! [M2: nat,N2: nat,G: nat > risk_Free_account,P3: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_942_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_943_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_944_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_945_add__nonpos__eq__0__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ Y @ zero_z1425366712893667068ccount )
=> ( ( ( plus_p1863581527469039996ccount @ X @ Y )
= zero_z1425366712893667068ccount )
= ( ( X = zero_z1425366712893667068ccount )
& ( Y = zero_z1425366712893667068ccount ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_946_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_947_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_948_add__nonneg__eq__0__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ X )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Y )
=> ( ( ( plus_p1863581527469039996ccount @ X @ Y )
= zero_z1425366712893667068ccount )
= ( ( X = zero_z1425366712893667068ccount )
& ( Y = zero_z1425366712893667068ccount ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_949_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_950_add__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 @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_951_add__nonpos__nonpos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_nonpos
thf(fact_952_add__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 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_953_add__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 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_954_add__nonneg__nonneg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_955_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_956_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_957_add__increasing2,axiom,
! [C: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ C )
=> ( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ord_le4245800335709223507ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_958_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_959_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_960_add__decreasing2,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ C @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_961_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_962_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_963_add__increasing,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_964_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_965_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_966_add__decreasing,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_967_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_968_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_969_add__less__le__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_970_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_971_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_972_add__le__less__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_973_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_974_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_real @ I @ J2 )
& ( ord_less_eq_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_975_add__mono__thms__linordered__field_I3_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J2 )
& ( ord_le4245800335709223507ccount @ K3 @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_976_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_977_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J2: real,K3: real,L: real] :
( ( ( ord_less_eq_real @ I @ J2 )
& ( ord_less_real @ K3 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K3 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_978_add__mono__thms__linordered__field_I4_J,axiom,
! [I: risk_Free_account,J2: risk_Free_account,K3: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J2 )
& ( ord_le2131251472502387783ccount @ K3 @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K3 ) @ ( plus_p1863581527469039996ccount @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_979_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_980_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_981_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_982_pos__add__strict,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_983_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_984_add__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 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_985_add__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 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_986_add__pos__pos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_987_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_988_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_989_add__neg__neg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_neg
thf(fact_990_add__le__imp__le__diff,axiom,
! [I: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_991_add__le__imp__le__diff,axiom,
! [I: real,K3: real,N2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ N2 )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_992_add__le__add__imp__diff__le,axiom,
! [I: nat,K3: nat,N2: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K3 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K3 ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_993_add__le__add__imp__diff__le,axiom,
! [I: real,K3: real,N2: real,J2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J2 @ K3 ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K3 ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J2 @ K3 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K3 ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_994_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_995_diff__le__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ C )
= ( ord_le4245800335709223507ccount @ A @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_996_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_997_le__diff__eq,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( minus_4846202936726426316ccount @ C @ B ) )
= ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_998_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_999_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1000_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1001_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1002_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1003_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1004_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1005_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1006_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1007_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1008_power__one__over,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_1009_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_1010_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1011_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1012_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1013_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1014_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1015_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1016_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1017_diff__less__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A @ B ) @ C )
= ( ord_le2131251472502387783ccount @ A @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1018_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1019_less__diff__eq,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( minus_4846202936726426316ccount @ C @ B ) )
= ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1020_eq__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1021_eq__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1022_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1023_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1024_update__account__subtract,axiom,
! [Rho: nat > real,I: real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F444380041991734328ccount @ Rho @ I @ ( minus_4846202936726426316ccount @ Alpha @ Beta ) )
= ( minus_4846202936726426316ccount @ ( risk_F444380041991734328ccount @ Rho @ I @ Alpha ) @ ( risk_F444380041991734328ccount @ Rho @ I @ Beta ) ) ) ).
% update_account_subtract
thf(fact_1025_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1026_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1027_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1028_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_1029_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_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 ) ) ) ) ).
% divide_le_cancel
thf(fact_1030_frac__less2,axiom,
! [X: real,Y: real,W: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_1031_frac__less,axiom,
! [X: real,Y: real,W: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_1032_frac__le,axiom,
! [Y: real,X: real,W: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_1033_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1034_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1035_mult__imp__less__div__pos,axiom,
! [Y: real,Z3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z3 @ Y ) @ X )
=> ( ord_less_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1036_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z3 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1037_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_1038_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1039_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1040_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_1041_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1042_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1043_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_1044_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_1045_add__divide__eq__if__simps_I4_J,axiom,
! [Z3: complex,A: complex,B: complex] :
( ( ( Z3 = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
= A ) )
& ( ( Z3 != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1046_add__divide__eq__if__simps_I4_J,axiom,
! [Z3: real,A: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= A ) )
& ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1047_diff__frac__eq,axiom,
! [Y: complex,Z3: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z3 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z3 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1048_diff__frac__eq,axiom,
! [Y: real,Z3: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1049_diff__divide__eq__iff,axiom,
! [Z3: complex,X: complex,Y: complex] :
( ( Z3 != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).
% diff_divide_eq_iff
thf(fact_1050_diff__divide__eq__iff,axiom,
! [Z3: real,X: real,Y: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).
% diff_divide_eq_iff
thf(fact_1051_divide__diff__eq__iff,axiom,
! [Z3: complex,X: complex,Y: complex] :
( ( Z3 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).
% divide_diff_eq_iff
thf(fact_1052_divide__diff__eq__iff,axiom,
! [Z3: real,X: real,Y: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).
% divide_diff_eq_iff
thf(fact_1053_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1054_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1055_add__strict__increasing2,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1056_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1057_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1058_add__strict__increasing,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1059_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1060_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1061_add__pos__nonneg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1062_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1063_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1064_add__nonpos__neg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_neg
thf(fact_1065_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1066_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1067_add__nonneg__pos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1068_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1069_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1070_add__neg__nonpos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_nonpos
thf(fact_1071_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1072_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1073_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1074_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_1075_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1076_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1077_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1078_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1079_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1080_less__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1081_less__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1082_square__diff__one__factored,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_1083_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1084_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1085_mult__imp__le__div__pos,axiom,
! [Y: real,Z3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ Y ) @ X )
=> ( ord_less_eq_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1086_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z3 @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1087_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_1088_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1089_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1090_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_1091_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_1092_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1093_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1094_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_1095_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_1096_frac__le__eq,axiom,
! [Y: real,Z3: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z3 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_1097_frac__less__eq,axiom,
! [Y: real,Z3: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z3 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_1098_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1099_power__diff,axiom,
! [A: complex,N2: nat,M2: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N2 ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_1100_power__diff,axiom,
! [A: real,N2: nat,M2: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N2 ) )
= ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_1101_power__diff,axiom,
! [A: nat,N2: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N2 ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_1102_loan__def,axiom,
( risk_Free_loan
= ( ^ [N3: nat,X2: real] :
( risk_F5458100604530014700ccount
@ ^ [M3: nat] : ( if_real @ ( N3 = M3 ) @ X2 @ zero_zero_real ) ) ) ) ).
% loan_def
thf(fact_1103_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_1104_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha2: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha2 @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_1105_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1106_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C4: real] :
( risk_F5458100604530014700ccount
@ ^ [N3: nat] : ( if_real @ ( N3 = zero_zero_nat ) @ C4 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_1107_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1108_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1109_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1110_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1111_bulk__update__algebraic__closed__form,axiom,
! [I: real,Rho: nat > real,N2: nat,Alpha: risk_Free_account] :
( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_nat @ N5 @ M4 )
=> ( ord_less_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( ( Rho @ zero_zero_nat )
= zero_zero_real )
=> ( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Alpha )
= ( plus_p1863581527469039996ccount
@ ( risk_Free_just_cash
@ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( risk_F1914734008469130493eserve @ Alpha ) )
@ ( groups6591440286371151544t_real
@ ^ [K5: nat] : ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( risk_F170160801229183585ccount @ Alpha @ K5 ) @ I ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( Rho @ K5 ) ) @ N2 ) ) ) @ ( plus_plus_real @ I @ ( Rho @ K5 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ ( risk_F4612863212915232279period @ Alpha ) ) ) ) )
@ ( groups6033208628184776703ccount
@ ^ [K5: nat] : ( risk_Free_loan @ K5 @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( Rho @ K5 ) ) @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha @ K5 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ ( risk_F4612863212915232279period @ Alpha ) ) ) ) ) ) ) ) ) ).
% bulk_update_algebraic_closed_form
thf(fact_1112_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1113_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1114_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1115_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1116_nat__add__left__cancel__less,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1117_nat__add__left__cancel__le,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1118_diff__diff__left,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% diff_diff_left
thf(fact_1119_bulk__update__account__zero,axiom,
! [N2: nat,Rho: nat > real,I: real] :
( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% bulk_update_account_zero
thf(fact_1120_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1121_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1122_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1123_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1124_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1125_Nat_Odiff__diff__right,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1126_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1127_div__le__mono,axiom,
! [M2: nat,N2: nat,K3: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K3 ) @ ( divide_divide_nat @ N2 @ K3 ) ) ) ).
% div_le_mono
thf(fact_1128_div__mult2__eq,axiom,
! [M2: nat,N2: nat,Q2: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N2 @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N2 ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_1129_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I3: nat,J: nat] :
( ( ( ord_less_nat @ J @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1130_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1131_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1132_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1133_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1134_add__lessD1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K3 )
=> ( ord_less_nat @ I @ K3 ) ) ).
% add_lessD1
thf(fact_1135_add__less__mono,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1136_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1137_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1138_add__less__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% add_less_mono1
thf(fact_1139_trans__less__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1140_trans__less__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1141_less__add__eq__less,axiom,
! [K3: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K3 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K3 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1142_add__leE,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K3 @ N2 ) ) ) ).
% add_leE
thf(fact_1143_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1144_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1145_add__leD1,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1146_add__leD2,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N2 )
=> ( ord_less_eq_nat @ K3 @ N2 ) ) ).
% add_leD2
thf(fact_1147_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K3 @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1148_add__le__mono,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1149_add__le__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% add_le_mono1
thf(fact_1150_trans__le__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1151_trans__le__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1152_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1153_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1154_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1155_diff__cancel2,axiom,
! [M2: nat,K3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K3 ) @ ( plus_plus_nat @ N2 @ K3 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1156_Nat_Odiff__cancel,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1157_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J2: nat,K3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib
thf(fact_1158_add__mult__distrib2,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K3 @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1159_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K3: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K3 )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% add_mult_distrib
thf(fact_1160_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1161_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1162_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1163_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1164_bulk__update__account_Osimps_I1_J,axiom,
! [Uu2: nat > real,Uv: real,Alpha: risk_Free_account] :
( ( risk_F2412532053715321062ccount @ zero_zero_nat @ Uu2 @ Uv @ Alpha )
= Alpha ) ).
% bulk_update_account.simps(1)
thf(fact_1165_bulk__update__account__plus,axiom,
! [N2: nat,Rho: nat > real,I: real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) )
= ( plus_p1863581527469039996ccount @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Alpha ) @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Beta ) ) ) ).
% bulk_update_account_plus
thf(fact_1166_div__le__mono2,axiom,
! [M2: nat,N2: nat,K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K3 @ N2 ) @ ( divide_divide_nat @ K3 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1167_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1168_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q2 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1169_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1170_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1171_bulk__update__account__subtract,axiom,
! [N2: nat,Rho: nat > real,I: real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ ( minus_4846202936726426316ccount @ Alpha @ Beta ) )
= ( minus_4846202936726426316ccount @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Alpha ) @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Beta ) ) ) ).
% bulk_update_account_subtract
thf(fact_1172_nat__mult__div__cancel__disj,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ( K3 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= zero_zero_nat ) )
& ( ( K3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1173_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1174_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K3: nat] :
( ! [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K3 ) @ ( F @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1175_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1176_less__diff__conv,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1177_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1178_le__diff__conv,axiom,
! [J2: nat,K3: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K3 ) ) ) ).
% le_diff_conv
thf(fact_1179_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1180_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K3 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1181_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1182_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K3 )
= ( J2
= ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1183_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1184_nat__mult__div__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1185_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1186_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1187_less__diff__conv2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_1188_nat__eq__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1189_nat__eq__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1190_nat__le__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1191_nat__le__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1192_nat__diff__add__eq1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1193_nat__diff__add__eq2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1194_nat__less__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1195_nat__less__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1196_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1197_bulk__update__account__mono,axiom,
! [I: real,Rho: nat > real,Alpha: risk_Free_account,Beta: risk_Free_account,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ord_less_eq_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( ord_le4245800335709223507ccount @ Alpha @ Beta )
=> ( ord_le4245800335709223507ccount @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Alpha ) @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Beta ) ) ) ) ) ) ).
% bulk_update_account_mono
thf(fact_1198_bulk__update__just__cash__closed__form,axiom,
! [Rho: nat > real,N2: nat,I: real,C: real] :
( ( ( Rho @ zero_zero_nat )
= zero_zero_real )
=> ( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ ( risk_Free_just_cash @ C ) )
= ( risk_Free_just_cash @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ C ) ) ) ) ).
% bulk_update_just_cash_closed_form
thf(fact_1199_bulk__update__loan__closed__form,axiom,
! [Rho: nat > real,K3: nat,I: real,N2: nat,C: real] :
( ( ( Rho @ K3 )
!= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( Rho @ K3 ) )
=> ( ( ( Rho @ zero_zero_nat )
= zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ ( risk_Free_loan @ K3 @ C ) )
= ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ C @ I ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( Rho @ K3 ) ) @ N2 ) ) ) @ ( plus_plus_real @ I @ ( Rho @ K3 ) ) ) ) @ ( risk_Free_loan @ K3 @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( Rho @ K3 ) ) @ N2 ) @ C ) ) ) ) ) ) ) ) ).
% bulk_update_loan_closed_form
thf(fact_1200_update__preserves__strictly__solvent,axiom,
! [I: real,Rho: nat > real,Alpha: risk_Free_account] :
( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ord_less_eq_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( risk_F1636578016437888323olvent @ Alpha )
=> ( risk_F1636578016437888323olvent @ ( risk_F444380041991734328ccount @ Rho @ I @ Alpha ) ) ) ) ) ) ).
% update_preserves_strictly_solvent
thf(fact_1201_additive__strictly__solvent,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_12 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_22 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_1202_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_1203_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_1204_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_1205_bulk__update__safety,axiom,
! [I: real,Rho: nat > real,Alpha: risk_Free_account,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ! [N5: nat] : ( ord_less_real @ ( Rho @ N5 ) @ one_one_real )
=> ( ! [N5: nat,M4: nat] :
( ( ord_less_eq_nat @ N5 @ M4 )
=> ( ord_less_eq_real @ ( Rho @ N5 ) @ ( Rho @ M4 ) ) )
=> ( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F2906766666041932210_value @ ( risk_F2412532053715321062ccount @ N2 @ Rho @ I @ Alpha ) ) ) ) ) ) ) ).
% bulk_update_safety
thf(fact_1206_net__asset__value__loan,axiom,
! [N2: nat,C: real] :
( ( risk_F2906766666041932210_value @ ( risk_Free_loan @ N2 @ C ) )
= C ) ).
% net_asset_value_loan
thf(fact_1207_net__asset__value__zero,axiom,
( ( risk_F2906766666041932210_value @ zero_z1425366712893667068ccount )
= zero_zero_real ) ).
% net_asset_value_zero
thf(fact_1208_net__asset__value__just__cash__left__inverse,axiom,
! [C: real] :
( ( risk_F2906766666041932210_value @ ( risk_Free_just_cash @ C ) )
= C ) ).
% net_asset_value_just_cash_left_inverse
thf(fact_1209_net__asset__value__plus,axiom,
! [Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) )
= ( plus_plus_real @ ( risk_F2906766666041932210_value @ Alpha ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_plus
thf(fact_1210_net__asset__value__mono,axiom,
! [Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha @ Beta )
=> ( ord_less_eq_real @ ( risk_F2906766666041932210_value @ Alpha ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_mono
thf(fact_1211_net__asset__value__minus,axiom,
! [Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( minus_4846202936726426316ccount @ Alpha @ Beta ) )
= ( minus_minus_real @ ( risk_F2906766666041932210_value @ Alpha ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_minus
thf(fact_1212_strictly__solvent__net__asset__value,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F2906766666041932210_value @ Alpha ) ) ) ).
% strictly_solvent_net_asset_value
thf(fact_1213_distribute__interest__plus,axiom,
! [I: real,Alpha: risk_Free_account,Beta: risk_Free_account] :
( ( risk_Free_just_cash @ ( times_times_real @ I @ ( risk_F2906766666041932210_value @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) ) ) )
= ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ ( times_times_real @ I @ ( risk_F2906766666041932210_value @ Alpha ) ) ) @ ( risk_Free_just_cash @ ( times_times_real @ I @ ( risk_F2906766666041932210_value @ Beta ) ) ) ) ) ).
% distribute_interest_plus
thf(fact_1214_net__asset__value__def,axiom,
( risk_F2906766666041932210_value
= ( ^ [Alpha2: risk_Free_account] :
( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha2 )
@ ( collect_nat
@ ^ [I3: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I3 )
!= zero_zero_real ) ) ) ) ) ).
% net_asset_value_def
thf(fact_1215_update__account__def,axiom,
( risk_F444380041991734328ccount
= ( ^ [Rho2: nat > real,I3: real,Alpha2: risk_Free_account] : ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ ( times_times_real @ I3 @ ( risk_F2906766666041932210_value @ Alpha2 ) ) ) @ ( risk_F2121631595377017831_loans @ Rho2 @ Alpha2 ) ) ) ) ).
% update_account_def
thf(fact_1216_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1217_sum__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1218_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1219_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1220_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1221_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1222_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( P @ K5 )
& ( ord_less_nat @ K5 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1223_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1224_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1225_finite__account__support,axiom,
! [Alpha: risk_Free_account] :
( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I3 )
!= zero_zero_real ) ) ) ).
% finite_account_support
thf(fact_1226_finite__Collect__le__nat,axiom,
! [K3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K3 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1227_finite__Collect__less__nat,axiom,
! [K3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K3 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1228_finite__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1229_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1230_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1231_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1232_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1233_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1234_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_1235_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1236_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_1237_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1238_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1239_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1240_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1241_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1242_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1243_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1244_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1245_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1246_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1247_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_1248_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1249_diff__Suc__diff__eq2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1250_diff__Suc__diff__eq1,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1251_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_1252_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1253_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_1254_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1255_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1256_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1257_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1258_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1259_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1260_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N5 )
=> ( P @ M ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1261_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1262_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X3 @ Z2 ) ) )
=> ( ! [N5: nat] : ( R3 @ N5 @ ( suc @ N5 ) )
=> ( R3 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1263_Suc__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M2 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1264_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1265_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups6591440286371151544t_real
@ ^ [J: nat] : ( risk_F170160801229183585ccount @ ( risk_Free_loan @ J @ ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ J ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ J ) ) ) @ ka )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ m ) )
= ( times_times_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( rho @ ka ) ) @ n ) @ ( risk_F170160801229183585ccount @ alpha @ ka ) ) ) ).
%------------------------------------------------------------------------------