TPTP Problem File: SLH0412^1.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_01263_039940__5980000_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1373 ( 520 unt; 100 typ; 0 def)
% Number of atoms : 3910 (1229 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11157 ( 259 ~; 78 |; 223 &;8852 @)
% ( 0 <=>;1745 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 553 ( 553 >; 0 *; 0 +; 0 <<)
% Number of symbols : 94 ( 91 usr; 16 con; 0-5 aty)
% Number of variables : 3769 ( 306 ^;3418 !; 45 ?;3769 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:44.747
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (91)
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_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
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__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Risk____Free____Lending__Oaccount,type,
groups8516999891779824987ccount: ( real > risk_Free_account ) > set_real > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Complex__Ocomplex,type,
groups8331919209915413362omplex: ( a > complex ) > set_a > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Real__Oreal,type,
groups2740460157737275248a_real: ( a > real ) > set_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Risk____Free____Lending__Oaccount,type,
groups4655409347963886775ccount: ( a > risk_Free_account ) > set_a > risk_Free_account ).
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_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Complex__Ocomplex_M_Eo_J,type,
top_top_complex_o: complex > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
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__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_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ofinite_Obalanced_001tf__a,type,
risk_Free_balanced_a: ( a > risk_Free_account ) > real > $o ).
thf(sy_c_Risk__Free__Lending_Ofinite_Ototal__interest_001tf__a,type,
risk_F6264599348697727258rest_a: ( a > risk_Free_account ) > real > real ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: 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_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Risk__Free__Lending_Otransfer_001tf__a,type,
risk_Free_transfer_a: ( a > risk_Free_account ) > risk_Free_account > a > a > a > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oupdate__account,type,
risk_F444380041991734328ccount: ( nat > real ) > real > risk_Free_account > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oupdate__ledger_001tf__a,type,
risk_F1245088672346398815dger_a: ( nat > real ) > real > ( a > risk_Free_account ) > a > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Ovalid__transfer,type,
risk_F1023690899723030139ansfer: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060L_062,type,
l: a > risk_Free_account ).
thf(sy_v__092_060rho_062,type,
rho: nat > real ).
thf(sy_v_c,type,
c: real ).
thf(sy_v_i,type,
i: real ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1267)
thf(fact_0__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_1_assms_I1_J,axiom,
( ( rho @ zero_zero_nat )
= zero_zero_real ) ).
% assms(1)
thf(fact_2__092_060open_062balanced_A_Iupdate__ledger_A_092_060rho_062_Ai_A_092_060L_062_J_A_I_I1_A_L_Ai_J_A_K_Ac_J_092_060close_062,axiom,
risk_Free_balanced_a @ ( risk_F1245088672346398815dger_a @ rho @ i @ l ) @ ( times_times_real @ ( plus_plus_real @ one_one_real @ i ) @ c ) ).
% \<open>balanced (update_ledger \<rho> i \<L>) ((1 + i) * c)\<close>
thf(fact_3_assms_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ i ).
% assms(3)
thf(fact_4_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_5_assms_I2_J,axiom,
! [N: nat] : ( ord_less_real @ ( rho @ N ) @ one_one_real ) ).
% assms(2)
thf(fact_6_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [Uu: a] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_7_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [Uu: a] : zero_z1425366712893667068ccount
@ A )
= zero_z1425366712893667068ccount ) ).
% sum.neutral_const
thf(fact_8_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_9_sum_Oneutral__const,axiom,
! [A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_10_local_Obalanced__alt__def,axiom,
( risk_Free_balanced_a
= ( ^ [L: a > risk_Free_account,C: real] :
( ( ( groups2740460157737275248a_real
@ ^ [A2: a] : ( risk_F1914734008469130493eserve @ ( L @ A2 ) )
@ top_top_set_a )
= C )
& ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups2740460157737275248a_real
@ ^ [A2: a] : ( risk_F170160801229183585ccount @ ( L @ A2 ) @ N2 )
@ top_top_set_a )
= zero_zero_real ) ) ) ) ) ).
% local.balanced_alt_def
thf(fact_11_update__ledger__def,axiom,
( risk_F1245088672346398815dger_a
= ( ^ [Rho: nat > real,I: real,L: a > risk_Free_account,A2: a] : ( risk_F444380041991734328ccount @ Rho @ I @ ( L @ A2 ) ) ) ) ).
% update_ledger_def
thf(fact_12_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_13_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_15_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_16_sum_Oneutral,axiom,
! [A: set_a,G: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups2740460157737275248a_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_17_sum_Oneutral,axiom,
! [A: set_a,G: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups4655409347963886775ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.neutral
thf(fact_18_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_19_sum_Oneutral,axiom,
! [A: set_complex,G: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_20_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_21_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_22_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_23_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_24_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_25_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > complex,A: set_a] :
( ( ( groups8331919209915413362omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_26_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > real,A: set_a] :
( ( ( groups2740460157737275248a_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_27_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > risk_Free_account,A: set_a] :
( ( ( groups4655409347963886775ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_28_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_29_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_30_local_Ototal__interest__def,axiom,
( risk_F6264599348697727258rest_a
= ( ^ [L: a > risk_Free_account,I: real] :
( groups2740460157737275248a_real
@ ^ [A2: a] : ( times_times_real @ I @ ( risk_F2906766666041932210_value @ ( L @ A2 ) ) )
@ top_top_set_a ) ) ) ).
% local.total_interest_def
thf(fact_31_sum_Oswap,axiom,
! [G: a > a > real,B: set_a,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I: a] : ( groups2740460157737275248a_real @ ( G @ I ) @ B )
@ A )
= ( groups2740460157737275248a_real
@ ^ [J: a] :
( groups2740460157737275248a_real
@ ^ [I: a] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_32_sum_Oswap,axiom,
! [G: a > nat > real,B: set_nat,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I: a] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups2740460157737275248a_real
@ ^ [I: a] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_33_sum_Oswap,axiom,
! [G: a > a > risk_Free_account,B: set_a,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [I: a] : ( groups4655409347963886775ccount @ ( G @ I ) @ B )
@ A )
= ( groups4655409347963886775ccount
@ ^ [J: a] :
( groups4655409347963886775ccount
@ ^ [I: a] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_34_sum_Oswap,axiom,
! [G: nat > a > real,B: set_a,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups2740460157737275248a_real @ ( G @ I ) @ B )
@ A )
= ( groups2740460157737275248a_real
@ ^ [J: a] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_35_sum_Oswap,axiom,
! [G: nat > nat > real,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_36_sum_Oswap,axiom,
! [G: complex > complex > complex,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_37_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_38_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_40_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_41_dual__order_Orefl,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_42_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_43_dual__order_Orefl,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_44_dual__order_Orefl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_45_add__left__cancel,axiom,
! [A4: real,B2: real,C2: real] :
( ( ( plus_plus_real @ A4 @ B2 )
= ( plus_plus_real @ A4 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_46_add__left__cancel,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A4 @ B2 )
= ( plus_plus_nat @ A4 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_47_add__left__cancel,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A4 @ B2 )
= ( plus_p1863581527469039996ccount @ A4 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_48_add__right__cancel,axiom,
! [B2: real,A4: real,C2: real] :
( ( ( plus_plus_real @ B2 @ A4 )
= ( plus_plus_real @ C2 @ A4 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_49_add__right__cancel,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A4 )
= ( plus_plus_nat @ C2 @ A4 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_50_add__right__cancel,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B2 @ A4 )
= ( plus_p1863581527469039996ccount @ C2 @ A4 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_51_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_52_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_53_add__le__cancel__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ A4 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_54_add__le__cancel__left,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_55_add__le__cancel__left,axiom,
! [C2: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) )
= ( ord_le4245800335709223507ccount @ A4 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_56_add__le__cancel__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
= ( ord_less_eq_real @ A4 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_57_add__le__cancel__right,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_58_add__le__cancel__right,axiom,
! [A4: risk_Free_account,C2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) )
= ( ord_le4245800335709223507ccount @ A4 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_59_add__0,axiom,
! [A4: real] :
( ( plus_plus_real @ zero_zero_real @ A4 )
= A4 ) ).
% add_0
thf(fact_60_add__0,axiom,
! [A4: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A4 )
= A4 ) ).
% add_0
thf(fact_61_add__0,axiom,
! [A4: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A4 )
= A4 ) ).
% add_0
thf(fact_62_add__0,axiom,
! [A4: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A4 )
= A4 ) ).
% add_0
thf(fact_63_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_64_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_65_add__cancel__right__right,axiom,
! [A4: real,B2: real] :
( ( A4
= ( plus_plus_real @ A4 @ B2 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_66_add__cancel__right__right,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( plus_plus_nat @ A4 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_67_add__cancel__right__right,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( A4
= ( plus_p1863581527469039996ccount @ A4 @ B2 ) )
= ( B2 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_right
thf(fact_68_add__cancel__right__right,axiom,
! [A4: complex,B2: complex] :
( ( A4
= ( plus_plus_complex @ A4 @ B2 ) )
= ( B2 = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_69_add__cancel__right__left,axiom,
! [A4: real,B2: real] :
( ( A4
= ( plus_plus_real @ B2 @ A4 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_70_add__cancel__right__left,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( plus_plus_nat @ B2 @ A4 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_71_add__cancel__right__left,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( A4
= ( plus_p1863581527469039996ccount @ B2 @ A4 ) )
= ( B2 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_left
thf(fact_72_add__cancel__right__left,axiom,
! [A4: complex,B2: complex] :
( ( A4
= ( plus_plus_complex @ B2 @ A4 ) )
= ( B2 = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_73_add__cancel__left__right,axiom,
! [A4: real,B2: real] :
( ( ( plus_plus_real @ A4 @ B2 )
= A4 )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_74_add__cancel__left__right,axiom,
! [A4: nat,B2: nat] :
( ( ( plus_plus_nat @ A4 @ B2 )
= A4 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_75_add__cancel__left__right,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A4 @ B2 )
= A4 )
= ( B2 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_right
thf(fact_76_add__cancel__left__right,axiom,
! [A4: complex,B2: complex] :
( ( ( plus_plus_complex @ A4 @ B2 )
= A4 )
= ( B2 = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_77_add__cancel__left__left,axiom,
! [B2: real,A4: real] :
( ( ( plus_plus_real @ B2 @ A4 )
= A4 )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_78_add__cancel__left__left,axiom,
! [B2: nat,A4: nat] :
( ( ( plus_plus_nat @ B2 @ A4 )
= A4 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_79_add__cancel__left__left,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B2 @ A4 )
= A4 )
= ( B2 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_left
thf(fact_80_add__cancel__left__left,axiom,
! [B2: complex,A4: complex] :
( ( ( plus_plus_complex @ B2 @ A4 )
= A4 )
= ( B2 = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_81_double__zero__sym,axiom,
! [A4: real] :
( ( zero_zero_real
= ( plus_plus_real @ A4 @ A4 ) )
= ( A4 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_82_add_Oright__neutral,axiom,
! [A4: real] :
( ( plus_plus_real @ A4 @ zero_zero_real )
= A4 ) ).
% add.right_neutral
thf(fact_83_add_Oright__neutral,axiom,
! [A4: nat] :
( ( plus_plus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% add.right_neutral
thf(fact_84_add_Oright__neutral,axiom,
! [A4: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A4 @ zero_z1425366712893667068ccount )
= A4 ) ).
% add.right_neutral
thf(fact_85_add_Oright__neutral,axiom,
! [A4: complex] :
( ( plus_plus_complex @ A4 @ zero_zero_complex )
= A4 ) ).
% add.right_neutral
thf(fact_86_add__less__cancel__left,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_87_add__less__cancel__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) )
= ( ord_less_real @ A4 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_88_add__less__cancel__left,axiom,
! [C2: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) )
= ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_89_add__less__cancel__right,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_90_add__less__cancel__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
= ( ord_less_real @ A4 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_91_add__less__cancel__right,axiom,
! [A4: risk_Free_account,C2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) )
= ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_92_mult_Oright__neutral,axiom,
! [A4: complex] :
( ( times_times_complex @ A4 @ one_one_complex )
= A4 ) ).
% mult.right_neutral
thf(fact_93_mult_Oright__neutral,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ one_one_real )
= A4 ) ).
% mult.right_neutral
thf(fact_94_mult_Oright__neutral,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ one_one_nat )
= A4 ) ).
% mult.right_neutral
thf(fact_95_mult__1,axiom,
! [A4: complex] :
( ( times_times_complex @ one_one_complex @ A4 )
= A4 ) ).
% mult_1
thf(fact_96_mult__1,axiom,
! [A4: real] :
( ( times_times_real @ one_one_real @ A4 )
= A4 ) ).
% mult_1
thf(fact_97_mult__1,axiom,
! [A4: nat] :
( ( times_times_nat @ one_one_nat @ A4 )
= A4 ) ).
% mult_1
thf(fact_98_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_99_net__asset__value__zero,axiom,
( ( risk_F2906766666041932210_value @ zero_z1425366712893667068ccount )
= zero_zero_real ) ).
% net_asset_value_zero
thf(fact_100_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A4 @ A4 ) )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_101_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A4 @ A4 ) @ zero_zero_real )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_102_le__add__same__cancel2,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ ( plus_plus_real @ B2 @ A4 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_103_le__add__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ ( plus_plus_nat @ B2 @ A4 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_104_le__add__same__cancel2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( plus_p1863581527469039996ccount @ B2 @ A4 ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_105_mem__Collect__eq,axiom,
! [A4: real,P: real > $o] :
( ( member_real @ A4 @ ( collect_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A4: a,P: a > $o] :
( ( member_a @ A4 @ ( collect_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A4: complex,P: complex > $o] :
( ( member_complex @ A4 @ ( collect_complex @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_113_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_115_le__add__same__cancel1,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ ( plus_plus_real @ A4 @ B2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_116_le__add__same__cancel1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ ( plus_plus_nat @ A4 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_117_le__add__same__cancel1,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_118_add__le__same__cancel2,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A4 @ B2 ) @ B2 )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_119_add__le__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A4 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_120_add__le__same__cancel2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ B2 )
= ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel2
thf(fact_121_add__le__same__cancel1,axiom,
! [B2: real,A4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A4 ) @ B2 )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_122_add__le__same__cancel1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A4 ) @ B2 )
= ( ord_less_eq_nat @ A4 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_123_add__le__same__cancel1,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ B2 @ A4 ) @ B2 )
= ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel1
thf(fact_124_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A4 @ A4 ) )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_125_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A4: real] :
( ( ord_less_real @ ( plus_plus_real @ A4 @ A4 ) @ zero_zero_real )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_126_less__add__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ ( plus_plus_nat @ B2 @ A4 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_127_less__add__same__cancel2,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ ( plus_plus_real @ B2 @ A4 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_128_less__add__same__cancel2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( plus_p1863581527469039996ccount @ B2 @ A4 ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_129_less__add__same__cancel1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ ( plus_plus_nat @ A4 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_130_less__add__same__cancel1,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ ( plus_plus_real @ A4 @ B2 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_131_less__add__same__cancel1,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_132_add__less__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A4 @ B2 ) @ B2 )
= ( ord_less_nat @ A4 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_133_add__less__same__cancel2,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A4 @ B2 ) @ B2 )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_134_add__less__same__cancel2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ B2 )
= ( ord_le2131251472502387783ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel2
thf(fact_135_add__less__same__cancel1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A4 ) @ B2 )
= ( ord_less_nat @ A4 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_136_add__less__same__cancel1,axiom,
! [B2: real,A4: real] :
( ( ord_less_real @ ( plus_plus_real @ B2 @ A4 ) @ B2 )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_137_add__less__same__cancel1,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ B2 @ A4 ) @ B2 )
= ( ord_le2131251472502387783ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel1
thf(fact_138_local_Ofinite__Rep__account__ledger,axiom,
! [L2: a > risk_Free_account,A: set_a,N3: nat] :
( ( risk_F170160801229183585ccount @ ( groups4655409347963886775ccount @ L2 @ A ) @ N3 )
= ( groups2740460157737275248a_real
@ ^ [A2: a] : ( risk_F170160801229183585ccount @ ( L2 @ A2 ) @ N3 )
@ A ) ) ).
% local.finite_Rep_account_ledger
thf(fact_139_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A4: real,B2: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A4 @ B2 ) @ C2 )
= ( plus_plus_real @ A4 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_140_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C2 )
= ( plus_plus_nat @ A4 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ C2 )
= ( plus_p1863581527469039996ccount @ A4 @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_142_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_143_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_real @ K @ L3 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_144_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I2 @ J2 )
& ( ord_le2131251472502387783ccount @ K @ L3 ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_145_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_real @ K @ L3 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_146_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_147_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I2 @ J2 )
& ( ord_le2131251472502387783ccount @ K @ L3 ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_148_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L3 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_149_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_150_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I2 @ J2 )
& ( ord_le4245800335709223507ccount @ K @ L3 ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_151_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_152_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( I2 = J2 )
& ( ord_less_real @ K @ L3 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_153_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( I2 = J2 )
& ( ord_le2131251472502387783ccount @ K @ L3 ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_154_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_155_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_156_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_157_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A4 @ B2 ) @ C2 )
= ( times_times_real @ A4 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_158_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A4 @ B2 ) @ C2 )
= ( times_times_nat @ A4 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_159_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( I2 = J2 )
& ( K = L3 ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_160_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( I2 = J2 )
& ( K = L3 ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_161_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( I2 = J2 )
& ( K = L3 ) )
=> ( ( plus_p1863581527469039996ccount @ I2 @ K )
= ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_162_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_163_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_164_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I2 @ J2 )
& ( K = L3 ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_165_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( I2 = J2 )
& ( ord_less_eq_real @ K @ L3 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_167_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( I2 = J2 )
& ( ord_le4245800335709223507ccount @ K @ L3 ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_168_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J2: real,K: real,L3: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L3 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_169_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_170_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: risk_Free_account,J2: risk_Free_account,K: risk_Free_account,L3: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I2 @ J2 )
& ( ord_le4245800335709223507ccount @ K @ L3 ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I2 @ K ) @ ( plus_p1863581527469039996ccount @ J2 @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_171_group__cancel_Oadd1,axiom,
! [A: real,K: real,A4: real,B2: real] :
( ( A
= ( plus_plus_real @ K @ A4 ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_172_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A4: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A4 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_173_group__cancel_Oadd1,axiom,
! [A: risk_Free_account,K: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ K @ A4 ) )
=> ( ( plus_p1863581527469039996ccount @ A @ B2 )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_174_group__cancel_Oadd2,axiom,
! [B: real,K: real,B2: real,A4: real] :
( ( B
= ( plus_plus_real @ K @ B2 ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_175_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A4: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_176_group__cancel_Oadd2,axiom,
! [B: risk_Free_account,K: risk_Free_account,B2: risk_Free_account,A4: risk_Free_account] :
( ( B
= ( plus_p1863581527469039996ccount @ K @ B2 ) )
=> ( ( plus_p1863581527469039996ccount @ A4 @ B )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_177_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_178_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_179_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_180_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_181_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_182_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_183_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_184_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_185_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_186_nless__le,axiom,
! [A4: real,B2: real] :
( ( ~ ( ord_less_real @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_187_nless__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_188_nless__le,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A4 @ B2 ) )
= ( ~ ( ord_le4245800335709223507ccount @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_189_nless__le,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_190_nle__le,axiom,
! [A4: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A4 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_191_nle__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_192_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_193_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_194_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_195_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_196_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_197_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_198_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_199_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_200_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_201_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_202_ord__eq__le__trans,axiom,
! [A4: real,B2: real,C2: real] :
( ( A4 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( A4 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4 = B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( A4 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__le__eq__trans,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_207_ord__le__eq__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_208_ord__le__eq__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_209_ord__le__eq__trans,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_210_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_211_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_212_antisym__conv1,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ~ ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_213_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_214_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_215_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_216_antisym__conv2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ~ ( ord_le2131251472502387783ccount @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_217_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_218_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_219_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_220_order__antisym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_222_order_Oasym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order.asym
thf(fact_223_order_Oasym,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ~ ( ord_less_real @ B2 @ A4 ) ) ).
% order.asym
thf(fact_224_order_Oasym,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A4 ) ) ).
% order.asym
thf(fact_225_comm__monoid__mult__class_Omult__1,axiom,
! [A4: complex] :
( ( times_times_complex @ one_one_complex @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_226_comm__monoid__mult__class_Omult__1,axiom,
! [A4: real] :
( ( times_times_real @ one_one_real @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_227_comm__monoid__mult__class_Omult__1,axiom,
! [A4: nat] :
( ( times_times_nat @ one_one_nat @ A4 )
= A4 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_228_add_Oassoc,axiom,
! [A4: real,B2: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A4 @ B2 ) @ C2 )
= ( plus_plus_real @ A4 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_229_add_Oassoc,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C2 )
= ( plus_plus_nat @ A4 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_230_add_Oassoc,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ C2 )
= ( plus_p1863581527469039996ccount @ A4 @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_231_order_Otrans,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_232_order_Otrans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_233_order_Otrans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_234_order_Otrans,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_235_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_236_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_237_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z2 )
=> ( ord_le4245800335709223507ccount @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_238_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_239_ord__eq__less__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( A4 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A4 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_240_ord__eq__less__trans,axiom,
! [A4: real,B2: real,C2: real] :
( ( A4 = B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A4 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_241_ord__eq__less__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4 = B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ A4 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_242_ord__less__eq__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A4 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_243_ord__less__eq__trans,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_real @ A4 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_244_ord__less__eq__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le2131251472502387783ccount @ A4 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_245_less__induct,axiom,
! [P: nat > $o,A4: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A4 ) ) ).
% less_induct
thf(fact_246_add_Oleft__cancel,axiom,
! [A4: real,B2: real,C2: real] :
( ( ( plus_plus_real @ A4 @ B2 )
= ( plus_plus_real @ A4 @ C2 ) )
= ( B2 = C2 ) ) ).
% add.left_cancel
thf(fact_247_add_Oleft__cancel,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A4 @ B2 )
= ( plus_p1863581527469039996ccount @ A4 @ C2 ) )
= ( B2 = C2 ) ) ).
% add.left_cancel
thf(fact_248_mult_Oassoc,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A4 @ B2 ) @ C2 )
= ( times_times_real @ A4 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% mult.assoc
thf(fact_249_mult_Oassoc,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A4 @ B2 ) @ C2 )
= ( times_times_nat @ A4 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% mult.assoc
thf(fact_250_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_251_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_252_linorder__wlog,axiom,
! [P: real > real > $o,A4: real,B2: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_253_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_254_add_Oright__cancel,axiom,
! [B2: real,A4: real,C2: real] :
( ( ( plus_plus_real @ B2 @ A4 )
= ( plus_plus_real @ C2 @ A4 ) )
= ( B2 = C2 ) ) ).
% add.right_cancel
thf(fact_255_add_Oright__cancel,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B2 @ A4 )
= ( plus_p1863581527469039996ccount @ C2 @ A4 ) )
= ( B2 = C2 ) ) ).
% add.right_cancel
thf(fact_256_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_257_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_258_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_259_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_260_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A2 )
& ( ord_less_eq_real @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_261_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_262_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A2 )
& ( ord_le4245800335709223507ccount @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_263_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_264_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_265_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_266_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ~ ( ord_le4245800335709223507ccount @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_267_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_268_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_269_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_270_dual__order_Oantisym,axiom,
! [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
=> ( ( ord_less_eq_real @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_271_dual__order_Oantisym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_272_dual__order_Oantisym,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_273_dual__order_Oantisym,axiom,
! [B2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_274_dual__order_Oasym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_275_dual__order_Oasym,axiom,
! [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
=> ~ ( ord_less_real @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_276_dual__order_Oasym,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A4 )
=> ~ ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_277_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B4: real] : ( plus_plus_real @ B4 @ A2 ) ) ) ).
% add.commute
thf(fact_278_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B4: nat] : ( plus_plus_nat @ B4 @ A2 ) ) ) ).
% add.commute
thf(fact_279_add_Ocommute,axiom,
( plus_p1863581527469039996ccount
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] : ( plus_p1863581527469039996ccount @ B4 @ A2 ) ) ) ).
% add.commute
thf(fact_280_dual__order_Otrans,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_281_dual__order_Otrans,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_282_dual__order_Otrans,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C2 @ B2 )
=> ( ord_le4245800335709223507ccount @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_283_dual__order_Otrans,axiom,
! [B2: set_nat,A4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_284_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_285_dual__order_Oirrefl,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_286_dual__order_Oirrefl,axiom,
! [A4: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_287_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_288_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B4: real] : ( times_times_real @ B4 @ A2 ) ) ) ).
% mult.commute
thf(fact_289_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B4: nat] : ( times_times_nat @ B4 @ A2 ) ) ) ).
% mult.commute
thf(fact_290_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_291_linorder__less__wlog,axiom,
! [P: real > real > $o,A4: real,B2: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_292_order_Ostrict__trans,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A4 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_293_order_Ostrict__trans,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A4 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_294_order_Ostrict__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ A4 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_295_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B4: real] :
( ( ord_less_real @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_296_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_297_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_298_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A2 @ B4 )
| ( A2 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_299_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B4: real] :
( ( ord_less_eq_real @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_300_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_301_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_302_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( A2 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_303_order_Ostrict__trans1,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A4 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_304_order_Ostrict__trans1,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A4 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_305_order_Ostrict__trans1,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ A4 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_306_order_Ostrict__trans1,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_307_order_Ostrict__trans2,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_real @ A4 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_308_order_Ostrict__trans2,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A4 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_309_order_Ostrict__trans2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ A4 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_310_order_Ostrict__trans2,axiom,
! [A4: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_311_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B4: real] :
( ( ord_less_eq_real @ A2 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_312_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_313_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B4 )
& ~ ( ord_le4245800335709223507ccount @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_314_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_315_add_Oleft__commute,axiom,
! [B2: real,A4: real,C2: real] :
( ( plus_plus_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) )
= ( plus_plus_real @ A4 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_316_add_Oleft__commute,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) )
= ( plus_plus_nat @ A4 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_317_add_Oleft__commute,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) )
= ( plus_p1863581527469039996ccount @ A4 @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_318_add__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_319_add__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_320_add__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ C2 @ D )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_321_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_322_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_323_mult_Ocomm__neutral,axiom,
! [A4: complex] :
( ( times_times_complex @ A4 @ one_one_complex )
= A4 ) ).
% mult.comm_neutral
thf(fact_324_mult_Ocomm__neutral,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ one_one_real )
= A4 ) ).
% mult.comm_neutral
thf(fact_325_mult_Ocomm__neutral,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ one_one_nat )
= A4 ) ).
% mult.comm_neutral
thf(fact_326_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_327_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_328_mult_Oleft__commute,axiom,
! [B2: real,A4: real,C2: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A4 @ C2 ) )
= ( times_times_real @ A4 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_329_mult_Oleft__commute,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A4 @ C2 ) )
= ( times_times_nat @ A4 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_330_dual__order_Ostrict__trans,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_331_dual__order_Ostrict__trans,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_332_dual__order_Ostrict__trans,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C2 @ B2 )
=> ( ord_le2131251472502387783ccount @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_333_add__left__imp__eq,axiom,
! [A4: real,B2: real,C2: real] :
( ( ( plus_plus_real @ A4 @ B2 )
= ( plus_plus_real @ A4 @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_334_add__left__imp__eq,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A4 @ B2 )
= ( plus_plus_nat @ A4 @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_335_add__left__imp__eq,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A4 @ B2 )
= ( plus_p1863581527469039996ccount @ A4 @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_336_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A2: real] :
( ( ord_less_real @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_337_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_nat @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_338_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B4: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_339_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B4 @ A2 )
| ( A2 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_340_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A2: real] :
( ( ord_less_eq_real @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_341_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_342_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B4: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_343_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
& ( A2 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_344_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_345_order_Ostrict__implies__not__eq,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_346_order_Ostrict__implies__not__eq,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_347_dual__order_Ostrict__trans1,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_348_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_349_dual__order_Ostrict__trans1,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C2 @ B2 )
=> ( ord_le2131251472502387783ccount @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_350_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
=> ( ( ord_less_set_nat @ C2 @ B2 )
=> ( ord_less_set_nat @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_351_dual__order_Ostrict__trans2,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_352_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_353_dual__order_Ostrict__trans2,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C2 @ B2 )
=> ( ord_le2131251472502387783ccount @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_354_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B2 @ A4 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_set_nat @ C2 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_355_add__right__imp__eq,axiom,
! [B2: real,A4: real,C2: real] :
( ( ( plus_plus_real @ B2 @ A4 )
= ( plus_plus_real @ C2 @ A4 ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_356_add__right__imp__eq,axiom,
! [B2: nat,A4: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A4 )
= ( plus_plus_nat @ C2 @ A4 ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_357_add__right__imp__eq,axiom,
! [B2: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B2 @ A4 )
= ( plus_p1863581527469039996ccount @ C2 @ A4 ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_358_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A2: real] :
( ( ord_less_eq_real @ B4 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_359_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_360_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B4: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A2 )
& ~ ( ord_le4245800335709223507ccount @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_361_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
& ~ ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_362_add__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_363_add__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_364_add__left__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_365_add__neg__nonpos,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_366_add__neg__nonpos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_367_add__neg__nonpos,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B2 @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_nonpos
thf(fact_368_add__nonneg__pos,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_369_add__nonneg__pos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_370_add__nonneg__pos,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B2 )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_371_add__nonpos__neg,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_372_add__nonpos__neg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_373_add__nonpos__neg,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B2 @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_neg
thf(fact_374_add__pos__nonneg,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_375_add__pos__nonneg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_376_add__pos__nonneg,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B2 )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_377_order_Ostrict__implies__order,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ord_less_eq_real @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_378_order_Ostrict__implies__order,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_379_order_Ostrict__implies__order,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ord_le4245800335709223507ccount @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_380_order_Ostrict__implies__order,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A4 @ B2 )
=> ( ord_less_eq_set_nat @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_381_less__eqE,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A4 @ C3 ) ) ) ).
% less_eqE
thf(fact_382_add__right__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_383_add__right__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_384_add__right__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_385_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% le_iff_add
thf(fact_386_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_387_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_388_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_389_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( ord_less_eq_real @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_390_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ord_less_eq_nat @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_391_dual__order_Ostrict__implies__order,axiom,
! [B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A4 )
=> ( ord_le4245800335709223507ccount @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_392_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B2 @ A4 )
=> ( ord_less_eq_set_nat @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_393_add__strict__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_394_add__strict__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_395_add__strict__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ C2 @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_396_add__le__less__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_397_add__le__less__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_398_add__le__less__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ C2 @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_399_add__less__le__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_400_add__less__le__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_401_add__less__le__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ C2 @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_402_add__le__imp__le__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) )
=> ( ord_less_eq_real @ A4 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_403_add__le__imp__le__left,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_404_add__le__imp__le__left,axiom,
! [C2: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) )
=> ( ord_le4245800335709223507ccount @ A4 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_405_add__le__imp__le__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
=> ( ord_less_eq_real @ A4 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_406_add__le__imp__le__right,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_407_add__le__imp__le__right,axiom,
! [A4: risk_Free_account,C2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) )
=> ( ord_le4245800335709223507ccount @ A4 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_408_add__strict__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_409_add__strict__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_410_add__strict__left__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_411_add__strict__increasing,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_412_add__strict__increasing,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_413_add__strict__increasing,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_414_add__strict__right__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_415_add__strict__right__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_416_add__strict__right__mono,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_417_add__strict__increasing2,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_418_add__strict__increasing2,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_419_add__strict__increasing2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_420_add__less__imp__less__left,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A4 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_421_add__less__imp__less__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A4 ) @ ( plus_plus_real @ C2 @ B2 ) )
=> ( ord_less_real @ A4 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_422_add__less__imp__less__left,axiom,
! [C2: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C2 @ A4 ) @ ( plus_p1863581527469039996ccount @ C2 @ B2 ) )
=> ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_423_add__less__imp__less__right,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A4 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_424_add__less__imp__less__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A4 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
=> ( ord_less_real @ A4 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_425_add__less__imp__less__right,axiom,
! [A4: risk_Free_account,C2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ ( plus_p1863581527469039996ccount @ B2 @ C2 ) )
=> ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_426_antisym,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_427_antisym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_428_antisym,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_429_antisym,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_430_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_431_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_432_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_433_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B4: real] :
( ( ord_less_eq_real @ A2 @ B4 )
& ( ord_less_eq_real @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_434_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_435_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [A2: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B4 )
& ( ord_le4245800335709223507ccount @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_436_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_437_order__subst1,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_438_order__subst1,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_439_order__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_440_order__subst1,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_441_order__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_442_order__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_443_order__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_444_order__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_445_order__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_446_order__subst1,axiom,
! [A4: real,F: set_nat > real,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_447_order__subst2,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_448_order__subst2,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_449_order__subst2,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_450_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_451_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_452_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_453_order__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_454_order__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_455_order__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_456_order__subst2,axiom,
! [A4: real,B2: real,F: real > set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_457_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_458_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_459_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_460_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_461_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_462_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_463_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_464_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_465_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_466_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_467_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_468_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_469_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_470_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_471_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_472_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_473_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_474_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_475_ord__eq__le__subst,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_476_ord__eq__le__subst,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_477_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_478_ord__eq__le__subst,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_479_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_480_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_481_ord__eq__le__subst,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_482_ord__eq__le__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_483_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_484_ord__eq__le__subst,axiom,
! [A4: set_nat,F: real > set_nat,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_485_ord__le__eq__subst,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_486_ord__le__eq__subst,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_487_ord__le__eq__subst,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_488_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_489_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_490_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_491_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_492_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_493_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_494_ord__le__eq__subst,axiom,
! [A4: real,B2: real,F: real > set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_495_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_496_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_497_order__less__asym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_asym
thf(fact_498_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_499_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_500_order__less__asym_H,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_501_order__less__asym_H,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ~ ( ord_less_real @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_502_order__less__asym_H,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_503_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_504_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_505_order__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z2 )
=> ( ord_le2131251472502387783ccount @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_506_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_507_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_508_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_509_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_510_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_511_ord__eq__less__subst,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_512_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_513_ord__eq__less__subst,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_514_ord__eq__less__subst,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_515_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_516_ord__eq__less__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_517_ord__eq__less__subst,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_518_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_519_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_520_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_521_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_522_ord__less__eq__subst,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_523_ord__less__eq__subst,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_524_ord__less__eq__subst,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_525_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_526_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_527_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_528_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_529_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_530_order__less__imp__le,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_531_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_532_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_533_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_534_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_535_order__less__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_536_order__less__subst1,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_537_order__less__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_538_order__less__subst1,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_539_order__less__subst1,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_540_order__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_541_order__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_542_order__less__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_543_order__less__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_544_order__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_545_order__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_546_order__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_547_order__less__subst2,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_548_order__less__subst2,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_549_order__less__subst2,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_550_order__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_551_order__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_552_order__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_553_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_554_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_555_order__antisym__conv,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_556_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_557_order__le__neq__trans,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_real @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_558_order__le__neq__trans,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_559_order__le__neq__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_560_order__le__neq__trans,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_set_nat @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_561_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_562_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_563_order__less__not__sym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_564_order__neq__le__trans,axiom,
! [A4: real,B2: real] :
( ( A4 != B2 )
=> ( ( ord_less_eq_real @ A4 @ B2 )
=> ( ord_less_real @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_565_order__neq__le__trans,axiom,
! [A4: nat,B2: nat] :
( ( A4 != B2 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_566_order__neq__le__trans,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( A4 != B2 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ord_le2131251472502387783ccount @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_567_order__neq__le__trans,axiom,
! [A4: set_nat,B2: set_nat] :
( ( A4 != B2 )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ord_less_set_nat @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_568_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_569_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_570_order__le__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z2 )
=> ( ord_le2131251472502387783ccount @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_571_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_572_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_573_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_574_order__less__imp__triv,axiom,
! [X: risk_Free_account,Y: risk_Free_account,P: $o] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_575_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_576_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_577_order__less__le__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z2 )
=> ( ord_le2131251472502387783ccount @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_578_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_579_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_580_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_581_order__le__less__subst1,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_582_order__le__less__subst1,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_583_order__le__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_584_order__le__less__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_585_order__le__less__subst1,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_586_order__le__less__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_587_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_588_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_589_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_590_order__le__less__subst1,axiom,
! [A4: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_591_order__le__less__subst2,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_592_order__le__less__subst2,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_593_order__le__less__subst2,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_594_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_595_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_596_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_597_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_598_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_599_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_600_order__le__less__subst2,axiom,
! [A4: real,B2: real,F: real > set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_601_order__less__le__subst1,axiom,
! [A4: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_602_order__less__le__subst1,axiom,
! [A4: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_603_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B2: real,C2: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_604_order__less__le__subst1,axiom,
! [A4: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_605_order__less__le__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_606_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B2: nat,C2: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_607_order__less__le__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_608_order__less__le__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_609_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_610_order__less__le__subst1,axiom,
! [A4: set_nat,F: real > set_nat,B2: real,C2: real] :
( ( ord_less_set_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_611_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_612_order__less__le__subst2,axiom,
! [A4: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_613_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_614_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_615_order__less__le__subst2,axiom,
! [A4: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_616_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_617_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_618_order__less__le__subst2,axiom,
! [A4: real,B2: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_619_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_620_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_621_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_622_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_623_order__less__imp__not__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_624_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_625_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_626_order__less__imp__not__eq2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_627_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_628_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_629_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_630_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_631_order__le__imp__less__or__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_632_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_633_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_634_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_635_order__less__imp__not__less,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_636_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_637_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_638_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_639_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_640_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_641_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_642_add__nonpos__nonpos,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_643_add__nonpos__nonpos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_644_add__nonpos__nonpos,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B2 @ zero_z1425366712893667068ccount )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_nonpos
thf(fact_645_add__nonneg__nonneg,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_646_add__nonneg__nonneg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_647_add__nonneg__nonneg,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_648_add__increasing2,axiom,
! [C2: real,B2: real,A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B2 @ A4 )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_649_add__increasing2,axiom,
! [C2: nat,B2: nat,A4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_650_add__increasing2,axiom,
! [C2: risk_Free_account,B2: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ C2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ A4 )
=> ( ord_le4245800335709223507ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_651_add__decreasing2,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A4 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_652_add__decreasing2,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_653_add__decreasing2,axiom,
! [C2: risk_Free_account,A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ C2 @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B2 )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_654_add__increasing,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_655_add__increasing,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_656_add__increasing,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C2 )
=> ( ord_le4245800335709223507ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_657_add__decreasing,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_658_add__decreasing,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_659_add__decreasing,axiom,
! [A4: risk_Free_account,C2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ C2 @ B2 )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_660_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_661_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_662_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_663_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_664_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_665_top_Oextremum__uniqueI,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A4 )
=> ( A4 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_666_top_Oextremum__uniqueI,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
=> ( A4 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_667_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_668_top_Oextremum__unique,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A4 )
= ( A4 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_669_top_Oextremum__unique,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
= ( A4 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_670_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_671_top__greatest,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ top_top_set_a ) ).
% top_greatest
thf(fact_672_top__greatest,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ top_top_set_real ) ).
% top_greatest
thf(fact_673_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_674_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_675_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_676_sum__mono,axiom,
! [K2: set_a,F: a > nat,G: a > nat] :
( ! [I3: a] :
( ( member_a @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K2 ) @ ( groups6334556678337121940_a_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_677_sum__mono,axiom,
! [K2: set_real,F: real > risk_Free_account,G: real > risk_Free_account] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ K2 ) @ ( groups8516999891779824987ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_678_sum__mono,axiom,
! [K2: set_a,F: a > real,G: a > real] :
( ! [I3: a] :
( ( member_a @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ K2 ) @ ( groups2740460157737275248a_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_679_sum__mono,axiom,
! [K2: set_a,F: a > risk_Free_account,G: a > risk_Free_account] :
( ! [I3: a] :
( ( member_a @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups4655409347963886775ccount @ F @ K2 ) @ ( groups4655409347963886775ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_680_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_681_pos__add__strict,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_682_pos__add__strict,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A4 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_683_pos__add__strict,axiom,
! [A4: risk_Free_account,B2: risk_Free_account,C2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C2 )
=> ( ord_le2131251472502387783ccount @ B2 @ ( plus_p1863581527469039996ccount @ A4 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_684_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ! [C3: nat] :
( ( B2
= ( plus_plus_nat @ A4 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_685_add__pos__pos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A4 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_686_add__pos__pos,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A4 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_687_add__pos__pos,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A4 )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B2 )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_688_add__neg__neg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_689_add__neg__neg,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_690_add__neg__neg,axiom,
! [A4: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B2 @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A4 @ B2 ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_neg
thf(fact_691_sum__nonpos,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_692_sum__nonpos,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_693_sum__nonpos,axiom,
! [A: set_a,F: a > nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_694_sum__nonpos,axiom,
! [A: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_695_sum__nonpos,axiom,
! [A: set_a,F: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_696_sum__nonpos,axiom,
! [A: set_a,F: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups4655409347963886775ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_697_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_698_sum__nonneg,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_699_sum__nonneg,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_700_sum__nonneg,axiom,
! [A: set_a,F: a > nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_701_sum__nonneg,axiom,
! [A: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups8516999891779824987ccount @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_702_sum__nonneg,axiom,
! [A: set_a,F: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups2740460157737275248a_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_703_sum__nonneg,axiom,
! [A: set_a,F: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups4655409347963886775ccount @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_704_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_705_net__asset__value__def,axiom,
( risk_F2906766666041932210_value
= ( ^ [Alpha: risk_Free_account] :
( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha )
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real ) ) ) ) ) ).
% net_asset_value_def
thf(fact_706_add_Ogroup__left__neutral,axiom,
! [A4: real] :
( ( plus_plus_real @ zero_zero_real @ A4 )
= A4 ) ).
% add.group_left_neutral
thf(fact_707_add_Ogroup__left__neutral,axiom,
! [A4: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A4 )
= A4 ) ).
% add.group_left_neutral
thf(fact_708_add_Ogroup__left__neutral,axiom,
! [A4: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A4 )
= A4 ) ).
% add.group_left_neutral
thf(fact_709_add_Ocomm__neutral,axiom,
! [A4: real] :
( ( plus_plus_real @ A4 @ zero_zero_real )
= A4 ) ).
% add.comm_neutral
thf(fact_710_add_Ocomm__neutral,axiom,
! [A4: nat] :
( ( plus_plus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% add.comm_neutral
thf(fact_711_add_Ocomm__neutral,axiom,
! [A4: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A4 @ zero_z1425366712893667068ccount )
= A4 ) ).
% add.comm_neutral
thf(fact_712_add_Ocomm__neutral,axiom,
! [A4: complex] :
( ( plus_plus_complex @ A4 @ zero_zero_complex )
= A4 ) ).
% add.comm_neutral
thf(fact_713_comm__monoid__add__class_Oadd__0,axiom,
! [A4: real] :
( ( plus_plus_real @ zero_zero_real @ A4 )
= A4 ) ).
% comm_monoid_add_class.add_0
thf(fact_714_comm__monoid__add__class_Oadd__0,axiom,
! [A4: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A4 )
= A4 ) ).
% comm_monoid_add_class.add_0
thf(fact_715_comm__monoid__add__class_Oadd__0,axiom,
! [A4: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A4 )
= A4 ) ).
% comm_monoid_add_class.add_0
thf(fact_716_comm__monoid__add__class_Oadd__0,axiom,
! [A4: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A4 )
= A4 ) ).
% comm_monoid_add_class.add_0
thf(fact_717_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_718_gr__implies__not__zero,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_719_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_720_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_721_top_Onot__eq__extremum,axiom,
! [A4: set_a] :
( ( A4 != top_top_set_a )
= ( ord_less_set_a @ A4 @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_722_top_Onot__eq__extremum,axiom,
! [A4: set_real] :
( ( A4 != top_top_set_real )
= ( ord_less_set_real @ A4 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_723_top_Oextremum__strict,axiom,
! [A4: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A4 ) ).
% top.extremum_strict
thf(fact_724_top_Oextremum__strict,axiom,
! [A4: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A4 ) ).
% top.extremum_strict
thf(fact_725_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_726_sum_Odistrib,axiom,
! [G: a > real,H: a > real,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [X3: a] : ( plus_plus_real @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_real @ ( groups2740460157737275248a_real @ G @ A ) @ ( groups2740460157737275248a_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_727_sum_Odistrib,axiom,
! [G: a > risk_Free_account,H: a > risk_Free_account,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [X3: a] : ( plus_p1863581527469039996ccount @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_p1863581527469039996ccount @ ( groups4655409347963886775ccount @ G @ A ) @ ( groups4655409347963886775ccount @ H @ A ) ) ) ).
% sum.distrib
thf(fact_728_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( plus_plus_real @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A ) @ ( groups6591440286371151544t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_729_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X3: complex] : ( plus_plus_complex @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A ) @ ( groups7754918857620584856omplex @ H @ A ) ) ) ).
% sum.distrib
thf(fact_730_sum__product,axiom,
! [F: a > real,A: set_a,G: a > real,B: set_a] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A ) @ ( groups2740460157737275248a_real @ G @ B ) )
= ( groups2740460157737275248a_real
@ ^ [I: a] :
( groups2740460157737275248a_real
@ ^ [J: a] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_731_sum__product,axiom,
! [F: a > real,A: set_a,G: nat > real,B: set_nat] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B ) )
= ( groups2740460157737275248a_real
@ ^ [I: a] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_732_sum__product,axiom,
! [F: nat > real,A: set_nat,G: a > real,B: set_a] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups2740460157737275248a_real @ G @ B ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups2740460157737275248a_real
@ ^ [J: a] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_733_sum__product,axiom,
! [F: nat > real,A: set_nat,G: nat > real,B: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_734_sum__product,axiom,
! [F: complex > complex,A: set_complex,G: complex > complex,B: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ ( groups7754918857620584856omplex @ G @ B ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_735_sum__distrib__right,axiom,
! [F: a > real,A: set_a,R: real] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A ) @ R )
= ( groups2740460157737275248a_real
@ ^ [N2: a] : ( times_times_real @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_736_sum__distrib__right,axiom,
! [F: nat > real,A: set_nat,R: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_737_sum__distrib__right,axiom,
! [F: complex > complex,A: set_complex,R: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ R )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_738_sum__distrib__left,axiom,
! [R: real,F: a > real,A: set_a] :
( ( times_times_real @ R @ ( groups2740460157737275248a_real @ F @ A ) )
= ( groups2740460157737275248a_real
@ ^ [N2: a] : ( times_times_real @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_739_sum__distrib__left,axiom,
! [R: real,F: nat > real,A: set_nat] :
( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_740_sum__distrib__left,axiom,
! [R: complex,F: complex > complex,A: set_complex] :
( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A ) )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_741_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_742_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_743_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_744_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_745_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_746_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_747_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_748_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_749_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: a > real,J2: real > a,T: set_a,H: a > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_a @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups2740460157737275248a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_750_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: a > real,J2: real > a,T: set_a,H: a > risk_Free_account,G: real > risk_Free_account] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_a @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_751_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J2: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_752_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: complex > real,J2: real > complex,T: set_complex,H: complex > complex,G: real > complex] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_complex @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_753_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: complex > a,J2: a > complex,T: set_complex,H: complex > complex,G: a > complex] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_complex @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8331919209915413362omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_754_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: real > a,J2: a > real,T: set_real,H: real > real,G: a > real] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_755_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: a > a,J2: a > a,T: set_a,H: a > real,G: a > real] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_a @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups2740460157737275248a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_756_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: nat > a,J2: a > nat,T: set_nat,H: nat > real,G: a > real] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_757_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: real > a,J2: a > real,T: set_real,H: real > risk_Free_account,G: a > risk_Free_account] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4655409347963886775ccount @ G @ S )
= ( groups8516999891779824987ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_758_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: a > a,J2: a > a,T: set_a,H: a > risk_Free_account,G: a > risk_Free_account] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_a @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4655409347963886775ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_759_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > real,A: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_760_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > real,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_761_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_762_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > real,A: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_complex @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_763_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > a,A: set_a,H: a > complex,Gamma: complex > complex,Phi: a > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_complex @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8331919209915413362omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_764_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > a,A: set_a,H: a > real,Gamma: real > real,Phi: a > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_765_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > a,A: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_766_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > a,A: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_767_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > a,A: set_a,H: a > real,Gamma: real > risk_Free_account,Phi: a > risk_Free_account] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4655409347963886775ccount @ Phi @ A )
= ( groups8516999891779824987ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_768_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > a,A: set_a,H: a > a,Gamma: a > risk_Free_account,Phi: a > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4655409347963886775ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_769_sum_Oeq__general,axiom,
! [B: set_a,A: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_770_sum_Oeq__general,axiom,
! [B: set_a,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_771_sum_Oeq__general,axiom,
! [B: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_772_sum_Oeq__general,axiom,
! [B: set_complex,A: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_complex @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_773_sum_Oeq__general,axiom,
! [B: set_complex,A: set_a,H: a > complex,Gamma: complex > complex,Phi: a > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_complex @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8331919209915413362omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_774_sum_Oeq__general,axiom,
! [B: set_real,A: set_a,H: a > real,Gamma: real > real,Phi: a > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_775_sum_Oeq__general,axiom,
! [B: set_a,A: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_776_sum_Oeq__general,axiom,
! [B: set_nat,A: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_777_sum_Oeq__general,axiom,
! [B: set_real,A: set_a,H: a > real,Gamma: real > risk_Free_account,Phi: a > risk_Free_account] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4655409347963886775ccount @ Phi @ A )
= ( groups8516999891779824987ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_778_sum_Oeq__general,axiom,
! [B: set_a,A: set_a,H: a > a,Gamma: a > risk_Free_account,Phi: a > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4655409347963886775ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_779_sum_Ocong,axiom,
! [A: set_a,B: set_a,G: a > real,H: a > real] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ A )
= ( groups2740460157737275248a_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_780_sum_Ocong,axiom,
! [A: set_a,B: set_a,G: a > risk_Free_account,H: a > risk_Free_account] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4655409347963886775ccount @ G @ A )
= ( groups4655409347963886775ccount @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_781_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_782_sum_Ocong,axiom,
! [A: set_complex,B: set_complex,G: complex > complex,H: complex > complex] :
( ( A = B )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= ( groups7754918857620584856omplex @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_783_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X3: nat] : $true ) ) ).
% UNIV_def
thf(fact_784_UNIV__def,axiom,
( top_top_set_complex
= ( collect_complex
@ ^ [X3: complex] : $true ) ) ).
% UNIV_def
thf(fact_785_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X3: a] : $true ) ) ).
% UNIV_def
thf(fact_786_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X3: real] : $true ) ) ).
% UNIV_def
thf(fact_787_local_Ototal__interest__transfer,axiom,
! [L2: a > risk_Free_account,Tau: risk_Free_account,A4: a,B2: a,I2: real] :
( ( risk_F6264599348697727258rest_a @ ( risk_Free_transfer_a @ L2 @ Tau @ A4 @ B2 ) @ I2 )
= ( risk_F6264599348697727258rest_a @ L2 @ I2 ) ) ).
% local.total_interest_transfer
thf(fact_788_local_Osum__transfer__equiv,axiom,
! [L2: a > risk_Free_account,Tau: risk_Free_account,X: a,Y: a] :
( ( groups4655409347963886775ccount @ L2 @ top_top_set_a )
= ( groups4655409347963886775ccount @ ( risk_Free_transfer_a @ L2 @ Tau @ X @ Y ) @ top_top_set_a ) ) ).
% local.sum_transfer_equiv
thf(fact_789_local_Obalanced__transfer,axiom,
! [Tau: risk_Free_account,A4: a,B2: a] :
( risk_Free_balanced_a
= ( ^ [L: a > risk_Free_account] : ( risk_Free_balanced_a @ ( risk_Free_transfer_a @ L @ Tau @ A4 @ B2 ) ) ) ) ).
% local.balanced_transfer
thf(fact_790_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_791_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_792_mult__cancel__right2,axiom,
! [A4: complex,C2: complex] :
( ( ( times_times_complex @ A4 @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A4 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_793_mult__cancel__right2,axiom,
! [A4: real,C2: real] :
( ( ( times_times_real @ A4 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A4 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_794_mult__cancel__right1,axiom,
! [C2: complex,B2: complex] :
( ( C2
= ( times_times_complex @ B2 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_795_mult__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_796_mult__cancel__left2,axiom,
! [C2: complex,A4: complex] :
( ( ( times_times_complex @ C2 @ A4 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A4 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_797_mult__cancel__left2,axiom,
! [C2: real,A4: real] :
( ( ( times_times_real @ C2 @ A4 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A4 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_798_mult__cancel__left1,axiom,
! [C2: complex,B2: complex] :
( ( C2
= ( times_times_complex @ C2 @ B2 ) )
= ( ( C2 = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_799_mult__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_800_convex__bound__lt,axiom,
! [X: real,A4: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A4 )
=> ( ( ord_less_real @ Y @ A4 )
=> ( ( 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 ) ) @ A4 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_801_local_Obalanced__def,axiom,
( risk_Free_balanced_a
= ( ^ [L: a > risk_Free_account,C: real] :
( ( groups4655409347963886775ccount @ L @ top_top_set_a )
= ( risk_Free_just_cash @ C ) ) ) ) ).
% local.balanced_def
thf(fact_802_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_803_local_Ojust__cash__UNIV__is__balanced,axiom,
! [L2: a > risk_Free_account] :
( ! [A3: a] :
? [C4: real] :
( ( L2 @ A3 )
= ( risk_Free_just_cash @ C4 ) )
=> ? [X_1: real] : ( risk_Free_balanced_a @ L2 @ X_1 ) ) ).
% local.just_cash_UNIV_is_balanced
thf(fact_804_local_Ojust__cash__summation,axiom,
! [A: set_a,L2: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [C4: real] :
( ( L2 @ X2 )
= ( risk_Free_just_cash @ C4 ) ) )
=> ? [C3: real] :
( ( groups4655409347963886775ccount @ L2 @ A )
= ( risk_Free_just_cash @ C3 ) ) ) ).
% local.just_cash_summation
thf(fact_805_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_806_add__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_807_mult__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_808_mult__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_809_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_810_mult__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_811_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_812_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_813_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_814_mult__zero__left,axiom,
! [A4: complex] :
( ( times_times_complex @ zero_zero_complex @ A4 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_815_mult__zero__left,axiom,
! [A4: real] :
( ( times_times_real @ zero_zero_real @ A4 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_816_mult__zero__left,axiom,
! [A4: nat] :
( ( times_times_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_817_mult__zero__right,axiom,
! [A4: complex] :
( ( times_times_complex @ A4 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_818_mult__zero__right,axiom,
! [A4: real] :
( ( times_times_real @ A4 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_819_mult__zero__right,axiom,
! [A4: nat] :
( ( times_times_nat @ A4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_820_mult__eq__0__iff,axiom,
! [A4: complex,B2: complex] :
( ( ( times_times_complex @ A4 @ B2 )
= zero_zero_complex )
= ( ( A4 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_821_mult__eq__0__iff,axiom,
! [A4: real,B2: real] :
( ( ( times_times_real @ A4 @ B2 )
= zero_zero_real )
= ( ( A4 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_822_mult__eq__0__iff,axiom,
! [A4: nat,B2: nat] :
( ( ( times_times_nat @ A4 @ B2 )
= zero_zero_nat )
= ( ( A4 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_823_mult__cancel__left,axiom,
! [C2: complex,A4: complex,B2: complex] :
( ( ( times_times_complex @ C2 @ A4 )
= ( times_times_complex @ C2 @ B2 ) )
= ( ( C2 = zero_zero_complex )
| ( A4 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_824_mult__cancel__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ( times_times_real @ C2 @ A4 )
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( A4 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_825_mult__cancel__left,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ( times_times_nat @ C2 @ A4 )
= ( times_times_nat @ C2 @ B2 ) )
= ( ( C2 = zero_zero_nat )
| ( A4 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_826_mult__cancel__right,axiom,
! [A4: complex,C2: complex,B2: complex] :
( ( ( times_times_complex @ A4 @ C2 )
= ( times_times_complex @ B2 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A4 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_827_mult__cancel__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ( times_times_real @ A4 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A4 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_828_mult__cancel__right,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ( times_times_nat @ A4 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A4 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_829_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_830_add__gr__0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_831_nat__0__less__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_832_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_833_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_834_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_835_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_836_Rep__account__plus,axiom,
! [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_1 @ Alpha_2 ) )
= ( ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N2 ) ) ) ) ).
% Rep_account_plus
thf(fact_837_just__cash__plus,axiom,
! [A4: real,B2: real] :
( ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ A4 ) @ ( risk_Free_just_cash @ B2 ) )
= ( risk_Free_just_cash @ ( plus_plus_real @ A4 @ B2 ) ) ) ).
% just_cash_plus
thf(fact_838_Rep__account__just__cash,axiom,
! [C2: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C2 ) )
= ( ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_839_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_840_mult__eq__self__implies__10,axiom,
! [M2: nat,N3: nat] :
( ( M2
= ( times_times_nat @ M2 @ N3 ) )
=> ( ( N3 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_841_just__cash__embed,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B4: real] :
( ( risk_Free_just_cash @ A2 )
= ( risk_Free_just_cash @ B4 ) ) ) ) ).
% just_cash_embed
thf(fact_842_transfer__collapse,axiom,
! [L2: a > risk_Free_account,Tau: risk_Free_account,A4: a] :
( ( risk_Free_transfer_a @ L2 @ Tau @ A4 @ A4 )
= L2 ) ).
% transfer_collapse
thf(fact_843_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B4: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A2 ) @ ( risk_Free_just_cash @ B4 ) ) ) ) ).
% just_cash_order_embed
thf(fact_844_add__eq__self__zero,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= M2 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_845_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_846_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_847_less__add__eq__less,axiom,
! [K: nat,L3: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ K @ L3 )
=> ( ( ( plus_plus_nat @ M2 @ L3 )
= ( plus_plus_nat @ K @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_848_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_849_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_850_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_851_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_852_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_853_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ) ).
% add_less_mono
thf(fact_854_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_855_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_856_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_857_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_858_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_859_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_860_le__neq__implies__less,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( M2 != N3 )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_861_less__or__eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_862_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_863_less__imp__le__nat,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_864_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_865_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_866_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_867_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_868_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_869_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_870_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_871_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_872_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_873_net__asset__value__just__cash__left__inverse,axiom,
! [C2: real] :
( ( risk_F2906766666041932210_value @ ( risk_Free_just_cash @ C2 ) )
= C2 ) ).
% net_asset_value_just_cash_left_inverse
thf(fact_874_net__asset__value__plus,axiom,
! [Alpha2: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( plus_p1863581527469039996ccount @ Alpha2 @ Beta ) )
= ( plus_plus_real @ ( risk_F2906766666041932210_value @ Alpha2 ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_plus
thf(fact_875_net__asset__value__mono,axiom,
! [Alpha2: risk_Free_account,Beta: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha2 @ Beta )
=> ( ord_less_eq_real @ ( risk_F2906766666041932210_value @ Alpha2 ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_mono
thf(fact_876_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_877_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_878_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_879_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_880_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_881_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_882_less__not__refl2,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( M2 != N3 ) ) ).
% less_not_refl2
thf(fact_883_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_884_nat__neq__iff,axiom,
! [M2: nat,N3: nat] :
( ( M2 != N3 )
= ( ( ord_less_nat @ M2 @ N3 )
| ( ord_less_nat @ N3 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_885_mult__not__zero,axiom,
! [A4: complex,B2: complex] :
( ( ( times_times_complex @ A4 @ B2 )
!= zero_zero_complex )
=> ( ( A4 != zero_zero_complex )
& ( B2 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_886_mult__not__zero,axiom,
! [A4: real,B2: real] :
( ( ( times_times_real @ A4 @ B2 )
!= zero_zero_real )
=> ( ( A4 != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_887_mult__not__zero,axiom,
! [A4: nat,B2: nat] :
( ( ( times_times_nat @ A4 @ B2 )
!= zero_zero_nat )
=> ( ( A4 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_888_divisors__zero,axiom,
! [A4: complex,B2: complex] :
( ( ( times_times_complex @ A4 @ B2 )
= zero_zero_complex )
=> ( ( A4 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_889_divisors__zero,axiom,
! [A4: real,B2: real] :
( ( ( times_times_real @ A4 @ B2 )
= zero_zero_real )
=> ( ( A4 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_890_divisors__zero,axiom,
! [A4: nat,B2: nat] :
( ( ( times_times_nat @ A4 @ B2 )
= zero_zero_nat )
=> ( ( A4 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_891_no__zero__divisors,axiom,
! [A4: complex,B2: complex] :
( ( A4 != zero_zero_complex )
=> ( ( B2 != zero_zero_complex )
=> ( ( times_times_complex @ A4 @ B2 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_892_no__zero__divisors,axiom,
! [A4: real,B2: real] :
( ( A4 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A4 @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_893_no__zero__divisors,axiom,
! [A4: nat,B2: nat] :
( ( A4 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A4 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_894_mult__left__cancel,axiom,
! [C2: complex,A4: complex,B2: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A4 )
= ( times_times_complex @ C2 @ B2 ) )
= ( A4 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_895_mult__left__cancel,axiom,
! [C2: real,A4: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A4 )
= ( times_times_real @ C2 @ B2 ) )
= ( A4 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_896_mult__left__cancel,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A4 )
= ( times_times_nat @ C2 @ B2 ) )
= ( A4 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_897_mult__right__cancel,axiom,
! [C2: complex,A4: complex,B2: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A4 @ C2 )
= ( times_times_complex @ B2 @ C2 ) )
= ( A4 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_898_mult__right__cancel,axiom,
! [C2: real,A4: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A4 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( A4 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_899_mult__right__cancel,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A4 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( A4 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_900_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_901_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_902_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_903_combine__common__factor,axiom,
! [A4: real,E: real,B2: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A4 @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A4 @ B2 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_904_combine__common__factor,axiom,
! [A4: nat,E: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A4 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A4 @ B2 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_905_distrib__right,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A4 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% distrib_right
thf(fact_906_distrib__right,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% distrib_right
thf(fact_907_distrib__left,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ A4 @ ( plus_plus_real @ B2 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A4 @ B2 ) @ ( times_times_real @ A4 @ C2 ) ) ) ).
% distrib_left
thf(fact_908_distrib__left,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( times_times_nat @ A4 @ ( plus_plus_nat @ B2 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A4 @ B2 ) @ ( times_times_nat @ A4 @ C2 ) ) ) ).
% distrib_left
thf(fact_909_comm__semiring__class_Odistrib,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A4 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_910_comm__semiring__class_Odistrib,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_911_ring__class_Oring__distribs_I1_J,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ A4 @ ( plus_plus_real @ B2 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A4 @ B2 ) @ ( times_times_real @ A4 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_912_ring__class_Oring__distribs_I2_J,axiom,
! [A4: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A4 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_913_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_914_gr__implies__not0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_915_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_916_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_917_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_918_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_919_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_920_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_921_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_922_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_923_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_924_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_925_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_926_mult__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_927_mult__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_928_mult__mono_H,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_929_mult__mono_H,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_930_zero__le__square,axiom,
! [A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ A4 ) ) ).
% zero_le_square
thf(fact_931_split__mult__pos__le,axiom,
! [A4: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_932_mult__left__mono__neg,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_933_mult__nonpos__nonpos,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_934_mult__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_935_mult__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_936_mult__right__mono__neg,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_937_mult__right__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_938_mult__right__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_939_mult__le__0__iff,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_940_split__mult__neg__le,axiom,
! [A4: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_941_split__mult__neg__le,axiom,
! [A4: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_942_mult__nonneg__nonneg,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_943_mult__nonneg__nonneg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A4 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_944_mult__nonneg__nonpos,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_945_mult__nonneg__nonpos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_946_mult__nonpos__nonneg,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_947_mult__nonpos__nonneg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_948_mult__nonneg__nonpos2,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A4 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_949_mult__nonneg__nonpos2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_950_zero__le__mult__iff,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_951_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_952_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_953_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_954_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_955_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_956_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_957_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_958_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_959_mult__neg__neg,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_960_not__square__less__zero,axiom,
! [A4: real] :
~ ( ord_less_real @ ( times_times_real @ A4 @ A4 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_961_mult__less__0__iff,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_962_mult__neg__pos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_963_mult__neg__pos,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_964_mult__pos__neg,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_965_mult__pos__neg,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_966_mult__pos__pos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A4 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_967_mult__pos__pos,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_968_mult__pos__neg2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A4 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_969_mult__pos__neg2,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A4 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_970_zero__less__mult__iff,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_971_zero__less__mult__pos,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A4 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_972_zero__less__mult__pos,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A4 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_973_zero__less__mult__pos2,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_974_zero__less__mult__pos2,axiom,
! [B2: real,A4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_975_mult__less__cancel__left__neg,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ B2 @ A4 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_976_mult__less__cancel__left__pos,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ A4 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_977_mult__strict__left__mono__neg,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_978_mult__strict__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_979_mult__strict__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_980_mult__less__cancel__left__disj,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A4 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A4 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_981_mult__strict__right__mono__neg,axiom,
! [B2: real,A4: real,C2: real] :
( ( ord_less_real @ B2 @ A4 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_982_mult__strict__right__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_983_mult__strict__right__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_984_mult__less__cancel__right__disj,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A4 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A4 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_985_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_986_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A4: real,B2: real,C2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_987_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_988_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_989_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_990_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_991_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_992_less__1__mult,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N3 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N3 ) ) ) ) ).
% less_1_mult
thf(fact_993_less__1__mult,axiom,
! [M2: real,N3: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N3 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N3 ) ) ) ) ).
% less_1_mult
thf(fact_994_add__mono1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_995_add__mono1,axiom,
! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A4 @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_996_less__add__one,axiom,
! [A4: nat] : ( ord_less_nat @ A4 @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ).
% less_add_one
thf(fact_997_less__add__one,axiom,
! [A4: real] : ( ord_less_real @ A4 @ ( plus_plus_real @ A4 @ one_one_real ) ) ).
% less_add_one
thf(fact_998_mult__less__le__imp__less,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_999_mult__less__le__imp__less,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1000_mult__le__less__imp__less,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1001_mult__le__less__imp__less,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1002_mult__right__le__imp__le,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1003_mult__right__le__imp__le,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1004_mult__left__le__imp__le,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1005_mult__left__le__imp__le,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1006_mult__le__cancel__left__pos,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ A4 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1007_mult__le__cancel__left__neg,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A4 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1008_mult__less__cancel__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A4 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1009_mult__strict__mono_H,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1010_mult__strict__mono_H,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1011_mult__right__less__imp__less,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1012_mult__right__less__imp__less,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1013_mult__less__cancel__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A4 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1014_mult__strict__mono,axiom,
! [A4: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1015_mult__strict__mono,axiom,
! [A4: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A4 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1016_mult__left__less__imp__less,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1017_mult__left__less__imp__less,axiom,
! [C2: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A4 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1018_mult__le__cancel__right,axiom,
! [A4: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1019_mult__le__cancel__left,axiom,
! [C2: real,A4: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1020_mult__left__le,axiom,
! [C2: real,A4: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ A4 ) ) ) ).
% mult_left_le
thf(fact_1021_mult__left__le,axiom,
! [C2: nat,A4: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ C2 ) @ A4 ) ) ) ).
% mult_left_le
thf(fact_1022_mult__le__one,axiom,
! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A4 @ B2 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1023_mult__le__one,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A4 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1031_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1032_mult__less__cancel__right2,axiom,
! [A4: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A4 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1033_mult__less__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1034_mult__less__cancel__left2,axiom,
! [C2: real,A4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A4 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A4 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1035_mult__less__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1036_mult__le__cancel__right2,axiom,
! [A4: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A4 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1037_mult__le__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1038_mult__le__cancel__left2,axiom,
! [C2: real,A4: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A4 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A4 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1039_mult__le__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1040_convex__bound__le,axiom,
! [X: real,A4: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A4 )
=> ( ( ord_less_eq_real @ Y @ A4 )
=> ( ( 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 ) ) @ A4 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1041_local_Ostrictly__solvent__transfer,axiom,
! [L2: a > risk_Free_account,Tau: risk_Free_account,X: a,Y: a] :
( ( risk_F1636578016437888323olvent @ ( groups4655409347963886775ccount @ L2 @ top_top_set_a ) )
= ( risk_F1636578016437888323olvent @ ( groups4655409347963886775ccount @ ( risk_Free_transfer_a @ L2 @ Tau @ X @ Y ) @ top_top_set_a ) ) ) ).
% local.strictly_solvent_transfer
thf(fact_1042_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1043_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N3 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1044_update__account__def,axiom,
( risk_F444380041991734328ccount
= ( ^ [Rho: nat > real,I: real,Alpha: risk_Free_account] : ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ ( times_times_real @ I @ ( risk_F2906766666041932210_value @ Alpha ) ) ) @ ( risk_F2121631595377017831_loans @ Rho @ Alpha ) ) ) ) ).
% update_account_def
thf(fact_1045_double__eq__0__iff,axiom,
! [A4: real] :
( ( ( plus_plus_real @ A4 @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_1046_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_1047_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_1048_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_1049_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_1050_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_1051_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_1052_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_1053_nat__1__eq__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N3 ) )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1054_nat__mult__eq__1__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1055_add__leE,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N3 )
=> ~ ( ord_less_eq_nat @ K @ N3 ) ) ) ).
% add_leE
thf(fact_1056_le__add1,axiom,
! [N3: nat,M2: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M2 ) ) ).
% le_add1
thf(fact_1057_le__add2,axiom,
! [N3: nat,M2: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M2 @ N3 ) ) ).
% le_add2
thf(fact_1058_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1059_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1060_in__mono,axiom,
! [A: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_1061_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_1062_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_1063_subsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_1064_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_1065_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_1066_add__leD1,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% add_leD1
thf(fact_1067_add__leD2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ K @ N3 ) ) ).
% add_leD2
thf(fact_1068_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1069_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_1070_eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( M2 = N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% eq_imp_le
thf(fact_1071_le__Suc__ex,axiom,
! [K: nat,L3: nat] :
( ( ord_less_eq_nat @ K @ L3 )
=> ? [N4: nat] :
( L3
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1072_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1073_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_1074_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( member_real @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1075_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1076_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1077_le__antisym,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M2 )
=> ( M2 = N3 ) ) ) ).
% le_antisym
thf(fact_1078_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1079_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1080_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1081_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_1082_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A5 )
=> ( member_real @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1083_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1084_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1085_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ) ).
% add_le_mono
thf(fact_1086_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_1087_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1088_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L3 ) ) ) ) ).
% mult_le_mono
thf(fact_1089_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_1090_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1091_subset__trans,axiom,
! [A: set_nat,B: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C5 )
=> ( ord_less_eq_set_nat @ A @ C5 ) ) ) ).
% subset_trans
thf(fact_1092_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1093_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1094_nat__le__linear,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
| ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% nat_le_linear
thf(fact_1095_trans__le__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1096_trans__le__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1097_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_1098_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1099_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A5 )
@ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1100_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1101_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1102_add__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N3 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1103_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1104_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1105_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1106_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N3 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% add_mult_distrib2
thf(fact_1107_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1108_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1109_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C5: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C5 )
=> ( ord_less_set_nat @ A @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_1110_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1111_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C5 )
=> ( ord_less_set_nat @ A @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_1112_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1113_less__account__def,axiom,
( ord_le2131251472502387783ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha_12 @ Alpha_22 )
& ~ ( ord_le4245800335709223507ccount @ Alpha_22 @ Alpha_12 ) ) ) ) ).
% less_account_def
thf(fact_1114_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1115_additive__strictly__solvent,axiom,
! [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_1 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_2 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_1 @ Alpha_2 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_1116_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1117_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1118_Collect__subset,axiom,
! [A: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1119_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1120_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_1121_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X5 ) ).
% linordered_field_no_lb
thf(fact_1122_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_1123_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1124_strictly__solvent__just__cash__equiv,axiom,
! [C2: real] :
( ( risk_F1636578016437888323olvent @ ( risk_Free_just_cash @ C2 ) )
= ( ord_less_eq_real @ zero_zero_real @ C2 ) ) ).
% strictly_solvent_just_cash_equiv
thf(fact_1125_strictly__solvent__net__asset__value,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F2906766666041932210_value @ Alpha2 ) ) ) ).
% strictly_solvent_net_asset_value
thf(fact_1126_strictly__solvent__non__negative__cash,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha2 ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_1127_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( M2 = N3 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1128_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1129_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1130_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1131_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1132_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1133_psubsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1134_psubsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1135_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A5 )
@ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_1136_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_1137_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1138_mult__less__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1139_add__scale__eq__noteq,axiom,
! [R: complex,A4: complex,B2: complex,C2: complex,D: complex] :
( ( R != zero_zero_complex )
=> ( ( ( A4 = B2 )
& ( C2 != D ) )
=> ( ( plus_plus_complex @ A4 @ ( times_times_complex @ R @ C2 ) )
!= ( plus_plus_complex @ B2 @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1140_add__scale__eq__noteq,axiom,
! [R: real,A4: real,B2: real,C2: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A4 = B2 )
& ( C2 != D ) )
=> ( ( plus_plus_real @ A4 @ ( times_times_real @ R @ C2 ) )
!= ( plus_plus_real @ B2 @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1141_add__scale__eq__noteq,axiom,
! [R: nat,A4: nat,B2: nat,C2: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A4 = B2 )
& ( C2 != D ) )
=> ( ( plus_plus_nat @ A4 @ ( times_times_nat @ R @ C2 ) )
!= ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1142_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1143_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_1144_just__cash__valid__transfer,axiom,
! [C2: real,T2: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C2 ) @ ( risk_Free_just_cash @ T2 ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T2 )
& ( ord_less_eq_real @ T2 @ C2 ) ) ) ).
% just_cash_valid_transfer
thf(fact_1145_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau2 )
& ( ord_le4245800335709223507ccount @ Tau2 @ Alpha ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_1146_Euclid__induct,axiom,
! [P: nat > nat > $o,A4: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1147_nat__descend__induct,axiom,
! [N3: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N3 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1148_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha2 @ Tau )
=> ( risk_F1636578016437888323olvent @ Alpha2 ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_1149_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1150_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1151_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N2 ) ) ) ) ) ).
% plus_account_def
thf(fact_1152_Rep__account__return__loans,axiom,
! [Rho2: nat > real,Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( risk_F2121631595377017831_loans @ Rho2 @ Alpha2 ) )
= ( ^ [N2: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( Rho2 @ N2 ) ) @ ( risk_F170160801229183585ccount @ Alpha2 @ N2 ) ) ) ) ).
% Rep_account_return_loans
thf(fact_1153_nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1154_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1155_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1156_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_1157_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1158_return__loans__def,axiom,
( risk_F2121631595377017831_loans
= ( ^ [Rho: nat > real,Alpha: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( times_times_real @ ( minus_minus_real @ one_one_real @ ( Rho @ N2 ) ) @ ( risk_F170160801229183585ccount @ Alpha @ N2 ) ) ) ) ) ).
% return_loans_def
thf(fact_1159_realpow__pos__nth,axiom,
! [N3: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N3 )
= A4 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1160_realpow__pos__nth__unique,axiom,
! [N3: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N3 )
= A4 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N3 )
= A4 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1161_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_1162_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C: real] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_1163_Bolzano,axiom,
! [A4: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( ! [A3: real,B3: real,C3: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A3 @ C3 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A4 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B2 )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X2 )
& ( ord_less_eq_real @ X2 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D2 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1164_sum__nth__roots,axiom,
! [N3: nat,C2: complex] :
( ( ord_less_nat @ one_one_nat @ N3 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N3 )
= C2 ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1165_sum__roots__unity,axiom,
! [N3: nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N3 )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1166_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1167_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1168_diff__diff__cancel,axiom,
! [I2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1169_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1170_zero__less__diff,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M2 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% zero_less_diff
thf(fact_1171_diff__is__0__eq,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1172_diff__is__0__eq_H,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1173_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1174_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1175_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1176_just__cash__subtract,axiom,
! [A4: real,B2: real] :
( ( minus_4846202936726426316ccount @ ( risk_Free_just_cash @ A4 ) @ ( risk_Free_just_cash @ B2 ) )
= ( risk_Free_just_cash @ ( minus_minus_real @ A4 @ B2 ) ) ) ).
% just_cash_subtract
thf(fact_1177_eq__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M2 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1178_le__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1179_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1180_diff__le__mono,axiom,
! [M2: nat,N3: nat,L3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L3 ) @ ( minus_minus_nat @ N3 @ L3 ) ) ) ).
% diff_le_mono
thf(fact_1181_diff__le__self,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ).
% diff_le_self
thf(fact_1182_le__diff__iff_H,axiom,
! [A4: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A4 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1183_diff__le__mono2,axiom,
! [M2: nat,N3: nat,L3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N3 ) @ ( minus_minus_nat @ L3 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1184_diffs0__imp__equal,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M2 )
= zero_zero_nat )
=> ( M2 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1185_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1186_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_1187_diff__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1188_diff__add__inverse2,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N3 ) @ N3 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1189_diff__add__inverse,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ N3 )
= M2 ) ).
% diff_add_inverse
thf(fact_1190_diff__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_cancel2
thf(fact_1191_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_1192_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1193_diff__less__mono2,axiom,
! [M2: nat,N3: nat,L3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ord_less_nat @ M2 @ L3 )
=> ( ord_less_nat @ ( minus_minus_nat @ L3 @ N3 ) @ ( minus_minus_nat @ L3 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1194_diff__less,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1195_less__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M2 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1196_diff__less__mono,axiom,
! [A4: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1197_diff__add__0,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1198_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1199_add__diff__inverse__nat,axiom,
! [M2: nat,N3: nat] :
( ~ ( ord_less_nat @ M2 @ N3 )
=> ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M2 @ N3 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1200_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1201_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1202_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1203_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1204_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1205_net__asset__value__minus,axiom,
! [Alpha2: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( minus_4846202936726426316ccount @ Alpha2 @ Beta ) )
= ( minus_minus_real @ ( risk_F2906766666041932210_value @ Alpha2 ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_minus
thf(fact_1206_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau2 )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha @ Tau2 ) ) ) ) ) ).
% valid_transfer_def
thf(fact_1207_nat__diff__split__asm,axiom,
! [P: nat > $o,A4: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A4 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A4 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A4
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1208_nat__diff__split,axiom,
! [P: nat > $o,A4: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A4 @ B2 ) )
= ( ( ( ord_less_nat @ A4 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A4
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1209_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1210_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N3 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1211_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N3 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1212_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N3 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1213_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N3 ) ) ) ).
% nat_le_add_iff1
thf(fact_1214_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N3 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1215_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 )
= N3 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1216_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N3 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1217_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N3 ) ) ) ).
% nat_less_add_iff1
thf(fact_1218_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1219_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z2: real] :
( ( X
= ( minus_minus_real @ Y @ Z2 ) )
= ( Y
= ( plus_plus_real @ X @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_1220_finite__atLeastAtMost,axiom,
! [L3: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L3 @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1221_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1222_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N3: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1223_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less_nat @ K4 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1224_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1225_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1226_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N3 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1227_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1228_finite__account__support,axiom,
! [Alpha2: risk_Free_account] :
( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I )
!= zero_zero_real ) ) ) ).
% finite_account_support
thf(fact_1229_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1230_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1231_local_Ofinite__UNIV,axiom,
finite_finite_a @ top_top_set_a ).
% local.finite_UNIV
thf(fact_1232_finite__nth__roots,axiom,
! [N3: nat,C2: complex] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N3 )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_1233_local_Ofinite,axiom,
! [A: set_a] : ( finite_finite_a @ A ) ).
% local.finite
thf(fact_1234_local_Ofinite__code,axiom,
( finite_finite_a
= ( ^ [A5: set_a] : $true ) ) ).
% local.finite_code
thf(fact_1235_all__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( P @ M ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1236_ex__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq_nat @ M @ N3 )
& ( P @ M ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1237_real__divide__square__eq,axiom,
! [R: real,A4: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A4 ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A4 @ R ) ) ).
% real_divide_square_eq
thf(fact_1238_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1239_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1240_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1241_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1242_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1243_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1244_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1245_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1246_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_1247_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1248_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1249_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( divide_divide_nat @ M2 @ N3 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1250_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1251_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1252_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1253_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1254_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1255_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) )
= ( divide_divide_nat @ M2 @ N3 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1256_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1257_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_1258_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1259_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1260_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1261_div__less,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( divide_divide_nat @ M2 @ N3 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1262_div__mult__self1__is__m,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N3 @ M2 ) @ N3 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1263_div__mult__self__is__m,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N3 ) @ N3 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1264_div__le__dividend,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N3 ) @ M2 ) ).
% div_le_dividend
thf(fact_1265_div__le__mono,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).
% div_le_mono
thf(fact_1266_div__mult2__eq,axiom,
! [M2: nat,N3: nat,Q2: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N3 @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N3 ) @ Q2 ) ) ).
% div_mult2_eq
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( zero_zero_real
= ( groups2740460157737275248a_real
@ ^ [A2: a] : ( risk_F170160801229183585ccount @ ( risk_F1245088672346398815dger_a @ rho @ i @ l @ A2 ) @ n )
@ top_top_set_a ) ) ).
%------------------------------------------------------------------------------