TPTP Problem File: SLH0256^1.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_00768_023587__5874534_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1432 ( 476 unt; 161 typ; 0 def)
% Number of atoms : 4083 (1131 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10628 ( 534 ~; 81 |; 310 &;7790 @)
% ( 0 <=>;1913 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 723 ( 723 >; 0 *; 0 +; 0 <<)
% Number of symbols : 145 ( 142 usr; 16 con; 0-3 aty)
% Number of variables : 3640 ( 357 ^;3062 !; 221 ?;3640 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:10.107
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_Ri1641125681238393385ccount: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
set_real_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Int__Oint_J_J,type,
set_real_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Real__Oreal_J_J,type,
set_int_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (142)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
finite8160569183818053390nt_int: set_int_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
finite3115048166472474290nt_nat: set_int_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Real__Oreal_J,type,
finite817158274246109198t_real: set_int_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
finite7161215471916998834at_int: set_nat_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
finite7853608736407863218t_real: set_nat_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Int__Oint_J,type,
finite8924082062276562062al_int: set_real_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
finite3878561044930982962al_nat: set_real_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
finite2300945044744945038l_real: set_real_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Risk____Free____Lending__Oaccount,type,
finite1362240334998357386ccount: set_Ri1641125681238393385ccount > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Risk____Free____Lending__Oaccount,type,
plus_p1863581527469039996ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
uminus_uminus_int_o: ( int > $o ) > int > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Risk____Free____Lending__Oaccount,type,
uminus3377898441596595772ccount: risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Risk____Free____Lending__Oaccount,type,
zero_z1425366712893667068ccount: 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_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices__Big_Olinorder_OMax_001t__Int__Oint,type,
lattices_Max_int: ( int > int > $o ) > set_int > int ).
thf(sy_c_Lattices__Big_Olinorder_OMax_001t__Nat__Onat,type,
lattices_Max_nat: ( nat > nat > $o ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder_OMax_001t__Real__Oreal,type,
lattices_Max_real: ( real > real > $o ) > set_real > real ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Int__Oint,type,
lattices_Min_int: ( int > int > $o ) > set_int > int ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Nat__Onat,type,
lattices_Min_nat: ( nat > nat > $o ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Real__Oreal,type,
lattices_Min_real: ( real > real > $o ) > set_real > real ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
lattic4275903605611617917x_real: set_real > real ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
lattic8718645017227715691in_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
lattic3629708407755379051n_real: set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Int__Oint,type,
lattic8443796201974363763nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Nat__Onat,type,
lattic8446286672483414039nt_nat: ( int > nat ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Risk____Free____Lending__Oaccount,type,
lattic7659584290714217530ccount: ( int > risk_Free_account ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Int__Oint,type,
lattic7444442490073309207at_int: ( nat > int ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Risk____Free____Lending__Oaccount,type,
lattic2248502109010755038ccount: ( nat > risk_Free_account ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Int__Oint,type,
lattic5053345968936924659al_int: ( real > int ) > set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
bot_bo4211708988250080445ccount: set_Ri1641125681238393385ccount ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_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_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Real__Vector__Spaces_Orepresentation_001t__Real__Oreal,type,
real_V2383402355066202452n_real: set_real > real > real > real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oshortest__period,type,
risk_F4612863212915232279period: risk_Free_account > nat ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Risk__Free__Lending_Ovalid__transfer,type,
risk_F1023690899723030139ansfer: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
collect_int_int: ( ( int > int ) > $o ) > set_int_int ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
collect_int_nat: ( ( int > nat ) > $o ) > set_int_nat ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Real__Oreal_J,type,
collect_int_real: ( ( int > real ) > $o ) > set_int_real ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
collect_nat_int: ( ( nat > int ) > $o ) > set_nat_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Int__Oint_J,type,
collect_real_int: ( ( real > int ) > $o ) > set_real_int ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
collect_real_nat: ( ( real > nat ) > $o ) > set_real_nat ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
collect_real_real: ( ( real > real ) > $o ) > set_real_real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Real__Oreal,type,
image_int_real: ( int > real ) > set_int > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: ( real > $o ) > real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Risk____Free____Lending__Oaccount,type,
member5612106785598075018ccount: risk_Free_account > set_Ri1641125681238393385ccount > $o ).
thf(sy_v__092_060alpha_062,type,
alpha: risk_Free_account ).
thf(sy_v_i,type,
i: nat ).
% Relevant facts (1263)
thf(fact_0_assms,axiom,
( ( risk_F170160801229183585ccount @ alpha @ i )
!= zero_zero_real ) ).
% assms
thf(fact_1_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_2__092_060open_062shortest__period_A_092_060alpha_062_A_092_060in_062_A_123i_O_A_092_060pi_062_A_092_060alpha_062_Ai_A_092_060noteq_062_A0_125_092_060close_062,axiom,
( member_nat @ ( risk_F4612863212915232279period @ alpha )
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I )
!= zero_zero_real ) ) ) ).
% \<open>shortest_period \<alpha> \<in> {i. \<pi> \<alpha> i \<noteq> 0}\<close>
thf(fact_3_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_4_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_5_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_6_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_7_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_8_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_9_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_10_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_11__092_060open_062shortest__period_A_092_060alpha_062_A_061_AMax_A_123i_O_A_092_060pi_062_A_092_060alpha_062_Ai_A_092_060noteq_062_A0_125_092_060close_062,axiom,
( ( risk_F4612863212915232279period @ alpha )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I )
!= zero_zero_real ) ) ) ) ).
% \<open>shortest_period \<alpha> = Max {i. \<pi> \<alpha> i \<noteq> 0}\<close>
thf(fact_12_B,axiom,
( ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I )
!= zero_zero_real ) )
!= bot_bot_set_nat ) ).
% B
thf(fact_13_A,axiom,
( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I )
!= zero_zero_real ) ) ) ).
% A
thf(fact_14_shortest__period__uminus,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F4612863212915232279period @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( risk_F4612863212915232279period @ Alpha ) ) ).
% shortest_period_uminus
thf(fact_15_representation__zero,axiom,
! [Basis: set_real] :
( ( real_V2383402355066202452n_real @ Basis @ zero_zero_real )
= ( ^ [B: real] : zero_zero_real ) ) ).
% representation_zero
thf(fact_16_add_Oinverse__inverse,axiom,
! [A: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_17_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_18_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_19_neg__equal__iff__equal,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= ( uminus3377898441596595772ccount @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_20_neg__equal__iff__equal,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_21_neg__equal__iff__equal,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_22_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_23_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_24_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_25_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_26_neg__equal__0__iff__equal,axiom,
! [A: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= zero_z1425366712893667068ccount )
= ( A = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_27_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_28_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_29_neg__0__equal__iff__equal,axiom,
! [A: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A ) )
= ( zero_z1425366712893667068ccount = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_30_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_31_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_32_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_33_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_34_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_35_equation__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A
= ( uminus3377898441596595772ccount @ B2 ) )
= ( B2
= ( uminus3377898441596595772ccount @ A ) ) ) ).
% equation_minus_iff
thf(fact_36_equation__minus__iff,axiom,
! [A: real,B2: real] :
( ( A
= ( uminus_uminus_real @ B2 ) )
= ( B2
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_37_equation__minus__iff,axiom,
! [A: int,B2: int] :
( ( A
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_38_minus__equation__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= B2 )
= ( ( uminus3377898441596595772ccount @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_39_minus__equation__iff,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= B2 )
= ( ( uminus_uminus_real @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_40_minus__equation__iff,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_41_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_42_finite__account__support,axiom,
! [Alpha: risk_Free_account] :
( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real ) ) ) ).
% finite_account_support
thf(fact_43_representation__ne__zero,axiom,
! [Basis: set_real,V: real,B2: real] :
( ( ( real_V2383402355066202452n_real @ Basis @ V @ B2 )
!= zero_zero_real )
=> ( member_real @ B2 @ Basis ) ) ).
% representation_ne_zero
thf(fact_44_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_45_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_46_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_47_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_48_Max__in,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( member_real @ ( lattic4275903605611617917x_real @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_49_Max__in,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( member_int @ ( lattic8263393255366662781ax_int @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_50_Max__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_51_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_52_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_53_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X2: real] :
~ ( member_real @ X2 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_54_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_55_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_56_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_57_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_58_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_59_verit__minus__simplify_I4_J,axiom,
! [B2: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_60_verit__minus__simplify_I4_J,axiom,
! [B2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_61_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_62_Rep__account__uminus,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N ) ) ) ) ).
% Rep_account_uminus
thf(fact_63_verit__negate__coefficient_I3_J,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A = B2 )
=> ( ( uminus3377898441596595772ccount @ A )
= ( uminus3377898441596595772ccount @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_64_verit__negate__coefficient_I3_J,axiom,
! [A: real,B2: real] :
( ( A = B2 )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_65_verit__negate__coefficient_I3_J,axiom,
! [A: int,B2: int] :
( ( A = B2 )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_66_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_67_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_68_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_69_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_70_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_71_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_72_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_80_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_81_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_82_empty__def,axiom,
( bot_bot_set_int
= ( collect_int
@ ^ [X2: int] : $false ) ) ).
% empty_def
thf(fact_83_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_84_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_85_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_86_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_87_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_88_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_89_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_90_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_91_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_92_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N ) ) ) ) ) ).
% uminus_account_def
thf(fact_93_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_94_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_95_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_96_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_97_shortest__period__def,axiom,
( risk_F4612863212915232279period
= ( ^ [Alpha2: risk_Free_account] :
( if_nat
@ ! [I: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I )
= zero_zero_real )
@ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I )
!= zero_zero_real ) ) ) ) ) ) ).
% shortest_period_def
thf(fact_98_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_99_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha2: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha2 @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_100_Max__const,axiom,
! [A2: set_int,C: nat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( lattic8265883725875713057ax_nat
@ ( image_int_nat
@ ^ [Uu: int] : C
@ A2 ) )
= C ) ) ) ).
% Max_const
thf(fact_101_Max__const,axiom,
! [A2: set_nat,C: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( image_nat_nat
@ ^ [Uu: nat] : C
@ A2 ) )
= C ) ) ) ).
% Max_const
thf(fact_102_Max_Obounded__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A2 ) @ X )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_103_Max_Obounded__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A2 ) @ X )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_104_Max_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_105_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_106_Max__less__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic4275903605611617917x_real @ A2 ) @ X )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_real @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_107_Max__less__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A2 ) @ X )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_108_Max__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_109_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_110_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_111_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_112_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_113_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_114_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_115_dual__order_Orefl,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% dual_order.refl
thf(fact_116_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_117_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_118_image__eqI,axiom,
! [B2: real,F: nat > real,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_119_image__eqI,axiom,
! [B2: nat,F: real > nat,X: real,A2: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_nat @ B2 @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_120_image__eqI,axiom,
! [B2: real,F: real > real,X: real,A2: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ B2 @ ( image_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_121_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_122_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_123_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_124_image__ident,axiom,
! [Y3: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_125_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_126_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_127_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_128_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_129_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_130_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_131_neg__le__iff__le,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_132_neg__le__iff__le,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le4245800335709223507ccount @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_133_neg__le__iff__le,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_134_neg__less__iff__less,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le2131251472502387783ccount @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_135_neg__less__iff__less,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_136_neg__less__iff__less,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_137_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_138_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_139_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_140_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_141_finite__imageI,axiom,
! [F2: set_nat,H: nat > int] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_142_finite__imageI,axiom,
! [F2: set_int,H: int > nat] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_143_finite__imageI,axiom,
! [F2: set_int,H: int > int] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_int @ ( image_int_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_144_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_145_neg__0__le__iff__le,axiom,
! [A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% neg_0_le_iff_le
thf(fact_146_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_147_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_148_neg__le__0__iff__le,axiom,
! [A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A ) @ zero_z1425366712893667068ccount )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A ) ) ).
% neg_le_0_iff_le
thf(fact_149_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_150_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_151_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_152_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_153_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_154_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_155_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_156_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_157_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_158_neg__0__less__iff__less,axiom,
! [A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% neg_0_less_iff_less
thf(fact_159_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_160_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_161_neg__less__0__iff__less,axiom,
! [A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A ) @ zero_z1425366712893667068ccount )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A ) ) ).
% neg_less_0_iff_less
thf(fact_162_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_163_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_164_just__cash__uminus,axiom,
! [A: real] :
( ( uminus3377898441596595772ccount @ ( risk_Free_just_cash @ A ) )
= ( risk_Free_just_cash @ ( uminus_uminus_real @ A ) ) ) ).
% just_cash_uminus
thf(fact_165_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_166_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_167_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_168_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_169_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_170_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_171_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B3
!= ( image_nat_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_172_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_173_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_174_rev__image__eqI,axiom,
! [X: real,A2: set_real,B2: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_175_rev__image__eqI,axiom,
! [X: real,A2: set_real,B2: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_176_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_177_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B3: set_real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_178_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B3: set_nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_179_image__subsetI,axiom,
! [A2: set_real,F: real > real,B3: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_180_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_181_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_182_image__cong,axiom,
! [M: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_183_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_184_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_185_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_186_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_187_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_188_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_189_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_190_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_191_verit__comp__simplify1_I1_J,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_192_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_193_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_194_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_195_verit__comp__simplify1_I2_J,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_196_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_197_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_198_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_199_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_200_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_201_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_202_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_203_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_204_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_205_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_206_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_207_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_208_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_209_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_210_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_211_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_212_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_213_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_214_nless__le,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A @ B2 ) )
= ( ~ ( ord_le4245800335709223507ccount @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_215_nless__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_int @ A @ B2 ) )
= ( ~ ( ord_less_eq_int @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_216_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_217_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_218_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_219_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_220_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_221_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_222_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_223_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_224_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_225_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_226_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_227_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_228_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_229_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z3: risk_Free_account] : ( Y4 = Z3 ) )
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_230_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_231_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_232_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_233_ord__eq__le__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( A = B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_235_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_236_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_237_ord__le__eq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_238_ord__le__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_239_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_240_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_241_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_242_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_243_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_244_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_245_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_246_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_247_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_248_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_249_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_250_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_251_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_252_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_253_order_Oasym,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A ) ) ).
% order.asym
thf(fact_254_order_Oasym,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order.asym
thf(fact_255_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_256_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_257_order_Otrans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% order.trans
thf(fact_258_order_Otrans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_259_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_260_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_261_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z )
=> ( ord_le4245800335709223507ccount @ X @ Z ) ) ) ).
% order_trans
thf(fact_262_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_263_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_264_ord__eq__less__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_265_ord__eq__less__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( A = B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_266_ord__eq__less__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_267_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_268_ord__less__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_269_ord__less__eq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_270_ord__less__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_271_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_272_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_273_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_274_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_275_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_276_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_277_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_278_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_279_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_280_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_281_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_282_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_283_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A3 )
& ( ord_less_eq_nat @ A3 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_284_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B: real] :
( ( ord_less_eq_real @ B @ A3 )
& ( ord_less_eq_real @ A3 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_285_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z3: risk_Free_account] : ( Y4 = Z3 ) )
= ( ^ [A3: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A3 )
& ( ord_le4245800335709223507ccount @ A3 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_286_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B: int] :
( ( ord_less_eq_int @ B @ A3 )
& ( ord_less_eq_int @ A3 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_287_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_288_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_289_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ~ ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_290_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_291_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_292_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_293_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_294_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_295_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_296_dual__order_Oantisym,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_297_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_298_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_299_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_300_dual__order_Oasym,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ~ ( ord_le2131251472502387783ccount @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_301_dual__order_Oasym,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_302_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_303_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_304_dual__order_Otrans,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B2 )
=> ( ord_le4245800335709223507ccount @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_305_dual__order_Otrans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_306_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_307_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_308_dual__order_Oirrefl,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% dual_order.irrefl
thf(fact_309_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_310_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_311_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_312_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A6: real,B6: real] :
( ( ord_less_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_313_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A6: int,B6: int] :
( ( ord_less_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int] : ( P @ A6 @ A6 )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_314_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_315_order_Ostrict__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_316_order_Ostrict__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_317_order_Ostrict__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_318_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
| ( A3 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_319_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
| ( A3 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_320_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A3 @ B )
| ( A3 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_321_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
| ( A3 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_322_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
& ( A3 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_323_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
& ( A3 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_324_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B )
& ( A3 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_325_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
& ( A3 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_326_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_327_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_328_order_Ostrict__trans1,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_329_order_Ostrict__trans1,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_330_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_331_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_332_order_Ostrict__trans2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_333_order_Ostrict__trans2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_334_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
& ~ ( ord_less_eq_nat @ B @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_335_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
& ~ ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_336_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B )
& ~ ( ord_le4245800335709223507ccount @ B @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_337_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
& ~ ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_338_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_339_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_340_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_341_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_342_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_343_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_344_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_345_dual__order_Ostrict__trans,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_346_dual__order_Ostrict__trans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_347_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
| ( A3 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_348_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
| ( A3 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_349_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B: risk_Free_account,A3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A3 )
| ( A3 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_350_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
| ( A3 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_351_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
& ( A3 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_352_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
& ( A3 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_353_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A3 )
& ( A3 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_354_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
& ( A3 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_355_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_356_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_357_order_Ostrict__implies__not__eq,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_358_order_Ostrict__implies__not__eq,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_359_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_360_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_361_dual__order_Ostrict__trans1,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_362_dual__order_Ostrict__trans1,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_363_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_364_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_365_dual__order_Ostrict__trans2,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_366_dual__order_Ostrict__trans2,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_367_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_368_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_369_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A3 )
& ~ ( ord_le4245800335709223507ccount @ A3 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_370_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_371_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_372_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_373_order_Ostrict__implies__order,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ord_le4245800335709223507ccount @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_374_order_Ostrict__implies__order,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_375_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_376_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_377_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_378_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_379_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_380_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_381_dual__order_Ostrict__implies__order,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ord_le4245800335709223507ccount @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_382_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_383_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_384_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_385_antisym,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_386_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_387_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
& ( ord_less_eq_nat @ B @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_388_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
& ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_389_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z3: risk_Free_account] : ( Y4 = Z3 ) )
= ( ^ [A3: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B )
& ( ord_le4245800335709223507ccount @ B @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_390_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
& ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_391_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_392_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_395_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_396_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_398_order__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_399_order__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_400_order__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_401_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_402_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_403_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_404_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_405_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_406_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_407_order__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_408_order__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_409_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_410_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_411_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_412_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_413_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_414_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_415_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_416_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_417_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_418_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_419_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_420_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_421_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_422_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_423_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_424_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_425_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_426_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_427_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_428_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_429_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_430_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_431_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_432_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_438_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_439_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_440_ord__eq__le__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_441_ord__eq__le__subst,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_442_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_443_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_444_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_446_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_452_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_453_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_454_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_455_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_456_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_457_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_458_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_459_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_460_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_461_order__less__asym_H,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_462_order__less__asym_H,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_463_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_464_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_465_order__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_466_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_467_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_468_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_469_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_470_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_471_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_472_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_473_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_474_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_475_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_476_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_477_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_478_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_479_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_480_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_481_ord__eq__less__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_482_ord__eq__less__subst,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_483_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_484_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_485_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_486_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_487_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_488_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_489_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_490_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_491_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_492_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_493_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_494_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_495_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_496_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_497_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_498_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_499_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_500_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_501_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_502_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_503_order__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_504_order__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_505_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_506_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_507_order__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_508_order__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_509_order__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_510_order__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_511_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_512_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_513_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_514_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_515_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_516_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_517_order__less__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_518_order__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_519_order__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_520_order__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_521_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_522_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_523_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_524_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_525_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_526_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_527_order__le__neq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_le2131251472502387783ccount @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_528_order__le__neq__trans,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_529_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_530_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_531_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_532_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_533_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_534_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_535_order__neq__le__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A != B2 )
=> ( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ord_le2131251472502387783ccount @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_536_order__neq__le__trans,axiom,
! [A: int,B2: int] :
( ( A != B2 )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_537_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_538_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_539_order__le__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_540_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_541_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_542_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_543_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_544_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_545_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_546_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_547_order__less__le__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_548_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_549_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_550_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_551_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_552_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_553_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_554_order__le__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_555_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_556_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_557_order__le__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_558_order__le__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_559_order__le__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_560_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_561_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_562_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_563_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_564_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_565_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_566_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_567_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_568_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_569_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_570_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_571_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_572_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_573_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_574_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_575_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_576_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_577_order__less__le__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_578_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_579_order__less__le__subst1,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_580_order__less__le__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_581_order__less__le__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_582_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_583_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_584_order__less__le__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_585_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_586_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_587_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_588_order__less__le__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_589_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_590_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_591_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_592_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_593_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_594_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_595_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_596_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_597_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_598_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_599_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_600_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_601_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_602_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_603_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_604_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_605_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_606_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_607_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_608_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_609_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_610_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_611_just__cash__embed,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B: real] :
( ( risk_Free_just_cash @ A3 )
= ( risk_Free_just_cash @ B ) ) ) ) ).
% just_cash_embed
thf(fact_612_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A3 ) @ ( risk_Free_just_cash @ B ) ) ) ) ).
% just_cash_order_embed
thf(fact_613_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_614_all__finite__subset__image,axiom,
! [F: int > nat,A2: set_int,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A2 ) )
=> ( P @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_615_all__finite__subset__image,axiom,
! [F: nat > int,A2: set_nat,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_616_all__finite__subset__image,axiom,
! [F: int > int,A2: set_int,P: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A2 ) )
=> ( P @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_617_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_618_ex__finite__subset__image,axiom,
! [F: int > nat,A2: set_int,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A2 )
& ( P @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_619_ex__finite__subset__image,axiom,
! [F: nat > int,A2: set_nat,P: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_620_ex__finite__subset__image,axiom,
! [F: int > int,A2: set_int,P: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A2 )
& ( P @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_621_finite__subset__image,axiom,
! [B3: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B3
= ( image_nat_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_622_finite__subset__image,axiom,
! [B3: set_nat,F: int > nat,A2: set_int] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A2 ) )
=> ? [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A2 )
& ( finite_finite_int @ C2 )
& ( B3
= ( image_int_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_623_finite__subset__image,axiom,
! [B3: set_int,F: nat > int,A2: set_nat] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A2 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( finite_finite_nat @ C2 )
& ( B3
= ( image_nat_int @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_624_finite__subset__image,axiom,
! [B3: set_int,F: int > int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A2 ) )
=> ? [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A2 )
& ( finite_finite_int @ C2 )
& ( B3
= ( image_int_int @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_625_finite__surj,axiom,
! [A2: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_626_finite__surj,axiom,
! [A2: set_nat,B3: set_int,F: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A2 ) )
=> ( finite_finite_int @ B3 ) ) ) ).
% finite_surj
thf(fact_627_finite__surj,axiom,
! [A2: set_int,B3: set_nat,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_628_finite__surj,axiom,
! [A2: set_int,B3: set_int,F: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A2 ) )
=> ( finite_finite_int @ B3 ) ) ) ).
% finite_surj
thf(fact_629_Compr__image__eq,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_630_Compr__image__eq,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_631_Compr__image__eq,axiom,
! [F: int > real,A2: set_int,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_int_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_real @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_632_Compr__image__eq,axiom,
! [F: real > nat,A2: set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_real_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_nat @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_633_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_634_Compr__image__eq,axiom,
! [F: int > nat,A2: set_int,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_nat @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_635_Compr__image__eq,axiom,
! [F: real > int,A2: set_real,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_real_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_int @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_636_Compr__image__eq,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_637_Compr__image__eq,axiom,
! [F: int > int,A2: set_int,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_int @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_638_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_639_imageE,axiom,
! [B2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_640_imageE,axiom,
! [B2: nat,F: real > nat,A2: set_real] :
( ( member_nat @ B2 @ ( image_real_nat @ F @ A2 ) )
=> ~ ! [X3: real] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_real @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_641_imageE,axiom,
! [B2: real,F: nat > real,A2: set_nat] :
( ( member_real @ B2 @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_642_imageE,axiom,
! [B2: real,F: real > real,A2: set_real] :
( ( member_real @ B2 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [X3: real] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_real @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_643_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_644_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_645_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_646_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_647_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_648_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_649_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_650_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_651_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_652_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_653_nat__seg__image__imp__finite,axiom,
! [A2: set_nat,F: nat > nat,N2: nat] :
( ( A2
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_654_nat__seg__image__imp__finite,axiom,
! [A2: set_int,F: nat > int,N2: nat] :
( ( A2
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_int @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_655_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
? [N: nat,F3: nat > nat] :
( A4
= ( image_nat_nat @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_656_finite__conv__nat__seg__image,axiom,
( finite_finite_int
= ( ^ [A4: set_int] :
? [N: nat,F3: nat > int] :
( A4
= ( image_nat_int @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_657_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_658_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_659_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_660_gr__implies__not__zero,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_661_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_662_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_663_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_664_less__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( uminus3377898441596595772ccount @ B2 ) )
= ( ord_le2131251472502387783ccount @ B2 @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% less_minus_iff
thf(fact_665_less__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_666_less__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_667_minus__less__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A ) @ B2 )
= ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_668_minus__less__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_669_minus__less__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_670_verit__negate__coefficient_I2_J,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_671_verit__negate__coefficient_I2_J,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_672_verit__negate__coefficient_I2_J,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_673_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_674_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_675_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ! [B7: set_int] :
( ( ord_less_set_int @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_676_le__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_677_le__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( uminus3377898441596595772ccount @ B2 ) )
= ( ord_le4245800335709223507ccount @ B2 @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% le_minus_iff
thf(fact_678_le__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_679_minus__le__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_680_minus__le__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A ) @ B2 )
= ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_681_minus__le__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_682_le__imp__neg__le,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_683_le__imp__neg__le,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% le_imp_neg_le
thf(fact_684_le__imp__neg__le,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_685_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_686_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_687_finite__has__maximal2,axiom,
! [A2: set_Ri1641125681238393385ccount,A: risk_Free_account] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( member5612106785598075018ccount @ A @ A2 )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ( ord_le4245800335709223507ccount @ A @ X3 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_688_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_689_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_690_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_691_finite__has__minimal2,axiom,
! [A2: set_Ri1641125681238393385ccount,A: risk_Free_account] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( member5612106785598075018ccount @ A @ A2 )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ( ord_le4245800335709223507ccount @ X3 @ A )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_692_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_693_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_694_finite__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_695_finite__subset,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_696_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_697_infinite__super,axiom,
! [S: set_int,T: set_int] :
( ( ord_less_eq_set_int @ S @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T ) ) ) ).
% infinite_super
thf(fact_698_rev__finite__subset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_699_rev__finite__subset,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_700_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_701_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_702_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > real] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_703_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_704_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > risk_Free_account] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S )
& ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ ( lattic7659584290714217530ccount @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_705_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > risk_Free_account] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ ( lattic2248502109010755038ccount @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_706_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic8443796201974363763nt_int @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_707_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_708_arg__min__least,axiom,
! [S: set_real,Y: real,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_709_arg__min__least,axiom,
! [S: set_int,Y: int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( member_int @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_710_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_711_arg__min__least,axiom,
! [S: set_real,Y: real,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_712_arg__min__least,axiom,
! [S: set_int,Y: int,F: int > real] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( member_int @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic2675449441010098035t_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_713_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_714_arg__min__least,axiom,
! [S: set_real,Y: real,F: real > int] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real @ Y @ S )
=> ( ord_less_eq_int @ ( F @ ( lattic5053345968936924659al_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_715_arg__min__least,axiom,
! [S: set_int,Y: int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( member_int @ Y @ S )
=> ( ord_less_eq_int @ ( F @ ( lattic8443796201974363763nt_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_716_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_int @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_717_pigeonhole__infinite,axiom,
! [A2: set_real,F: real > nat] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ ( image_real_nat @ F @ A2 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_718_pigeonhole__infinite,axiom,
! [A2: set_real,F: real > int] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ ( image_real_int @ F @ A2 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_719_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_720_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_721_pigeonhole__infinite,axiom,
! [A2: set_int,F: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_722_pigeonhole__infinite,axiom,
! [A2: set_int,F: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_723_Max__mono,axiom,
! [M: set_real,N3: set_real] :
( ( ord_less_eq_set_real @ M @ N3 )
=> ( ( M != bot_bot_set_real )
=> ( ( finite_finite_real @ N3 )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ M ) @ ( lattic4275903605611617917x_real @ N3 ) ) ) ) ) ).
% Max_mono
thf(fact_724_Max__mono,axiom,
! [M: set_int,N3: set_int] :
( ( ord_less_eq_set_int @ M @ N3 )
=> ( ( M != bot_bot_set_int )
=> ( ( finite_finite_int @ N3 )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M ) @ ( lattic8263393255366662781ax_int @ N3 ) ) ) ) ) ).
% Max_mono
thf(fact_725_Max__mono,axiom,
! [M: set_nat,N3: set_nat] :
( ( ord_less_eq_set_nat @ M @ N3 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N3 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N3 ) ) ) ) ) ).
% Max_mono
thf(fact_726_Max_Osubset__imp,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( finite_finite_real @ B3 )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A2 ) @ ( lattic4275903605611617917x_real @ B3 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_727_Max_Osubset__imp,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( finite_finite_int @ B3 )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A2 ) @ ( lattic8263393255366662781ax_int @ B3 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_728_Max_Osubset__imp,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_729_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_real,D: real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite2300945044744945038l_real
@ ( collect_real_real
@ ^ [F3: real > real] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_730_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_nat,D: nat] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite3878561044930982962al_nat
@ ( collect_real_nat
@ ^ [F3: real > nat] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_nat @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_731_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_int,D: int] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite8924082062276562062al_int
@ ( collect_real_int
@ ^ [F3: real > int] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_int @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_732_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_real,D: real] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite7853608736407863218t_real
@ ( collect_nat_real
@ ^ [F3: nat > real] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_real @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_733_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_nat,D: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [F3: nat > nat] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_734_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_int,D: int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite7161215471916998834at_int
@ ( collect_nat_int
@ ^ [F3: nat > int] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_int @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_735_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_real,D: real] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite817158274246109198t_real
@ ( collect_int_real
@ ^ [F3: int > real] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_real @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_736_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_nat,D: nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite3115048166472474290nt_nat
@ ( collect_int_nat
@ ^ [F3: int > nat] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_nat @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_737_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_int,D: int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite8160569183818053390nt_int
@ ( collect_int_int
@ ^ [F3: int > int] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_int @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F3 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_738_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_739_infinite__growing,axiom,
! [X6: set_real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X6 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X6 ) ) ) ).
% infinite_growing
thf(fact_740_infinite__growing,axiom,
! [X6: set_int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X6 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X6 ) ) ) ).
% infinite_growing
thf(fact_741_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_742_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_743_ex__min__if__finite,axiom,
! [S: set_Ri1641125681238393385ccount] :
( ( finite1362240334998357386ccount @ S )
=> ( ( S != bot_bo4211708988250080445ccount )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ S )
& ~ ? [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ S )
& ( ord_le2131251472502387783ccount @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_744_ex__min__if__finite,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_745_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_746_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_747_finite__has__maximal,axiom,
! [A2: set_Ri1641125681238393385ccount] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( A2 != bot_bo4211708988250080445ccount )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_748_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_749_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_750_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_751_finite__has__minimal,axiom,
! [A2: set_Ri1641125681238393385ccount] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( A2 != bot_bo4211708988250080445ccount )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_752_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_753_Max__ge,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A2 ) ) ) ) ).
% Max_ge
thf(fact_754_Max__ge,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A2 ) ) ) ) ).
% Max_ge
thf(fact_755_Max__ge,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max_ge
thf(fact_756_Max__eqI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ A2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( ( lattic4275903605611617917x_real @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_757_Max__eqI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( member_int @ X @ A2 )
=> ( ( lattic8263393255366662781ax_int @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_758_Max__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_759_Max__eq__if,axiom,
! [A2: set_real,B3: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B3 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B3 )
=> ? [Xa: real] :
( ( member_real @ Xa @ A2 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ( lattic4275903605611617917x_real @ A2 )
= ( lattic4275903605611617917x_real @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_760_Max__eq__if,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ A2 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ( lattic8263393255366662781ax_int @ A2 )
= ( lattic8263393255366662781ax_int @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_761_Max__eq__if,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A2 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_762_Max_OcoboundedI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ord_less_eq_real @ A @ ( lattic4275903605611617917x_real @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_763_Max_OcoboundedI,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ( ord_less_eq_int @ A @ ( lattic8263393255366662781ax_int @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_764_Max_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_765_Max__gr__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic4275903605611617917x_real @ A2 ) )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_real @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_766_Max__gr__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_767_Max__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_768_Max__eq__iff,axiom,
! [A2: set_real,M3: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ( lattic4275903605611617917x_real @ A2 )
= M3 )
= ( ( member_real @ M3 @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ X2 @ M3 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_769_Max__eq__iff,axiom,
! [A2: set_int,M3: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ( lattic8263393255366662781ax_int @ A2 )
= M3 )
= ( ( member_int @ M3 @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X2 @ M3 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_770_Max__eq__iff,axiom,
! [A2: set_nat,M3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A2 )
= M3 )
= ( ( member_nat @ M3 @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_771_Max__ge__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A2 ) )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_772_Max__ge__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_773_Max__ge__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_774_eq__Max__iff,axiom,
! [A2: set_real,M3: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( M3
= ( lattic4275903605611617917x_real @ A2 ) )
= ( ( member_real @ M3 @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ X2 @ M3 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_775_eq__Max__iff,axiom,
! [A2: set_int,M3: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( M3
= ( lattic8263393255366662781ax_int @ A2 ) )
= ( ( member_int @ M3 @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X2 @ M3 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_776_eq__Max__iff,axiom,
! [A2: set_nat,M3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M3
= ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ( member_nat @ M3 @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_777_Max_OboundedE,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A2 ) @ X )
=> ! [A8: real] :
( ( member_real @ A8 @ A2 )
=> ( ord_less_eq_real @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_778_Max_OboundedE,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A2 ) @ X )
=> ! [A8: int] :
( ( member_int @ A8 @ A2 )
=> ( ord_less_eq_int @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_779_Max_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_780_Max_OboundedI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [A6: real] :
( ( member_real @ A6 @ A2 )
=> ( ord_less_eq_real @ A6 @ X ) )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_781_Max_OboundedI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ord_less_eq_int @ A6 @ X ) )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_782_Max_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_783_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_784_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C3: real] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C3 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_785_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_786_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_787_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_788_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_789_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_790_dual__Min,axiom,
( ( lattices_Min_real
@ ^ [X2: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X2 ) )
= lattic4275903605611617917x_real ) ).
% dual_Min
thf(fact_791_dual__Min,axiom,
( ( lattices_Min_int
@ ^ [X2: int,Y5: int] : ( ord_less_eq_int @ Y5 @ X2 ) )
= lattic8263393255366662781ax_int ) ).
% dual_Min
thf(fact_792_dual__Min,axiom,
( ( lattices_Min_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 ) )
= lattic8265883725875713057ax_nat ) ).
% dual_Min
thf(fact_793_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_794_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_795_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_796_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_797_minf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_798_minf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_799_minf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_eq_int @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_800_minf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_801_minf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_eq_real @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_802_minf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_eq_int @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_803_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_804_subsetI,axiom,
! [A2: set_real,B3: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B3 ) )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_805_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_806_ComplI,axiom,
! [C: real,A2: set_real] :
( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% ComplI
thf(fact_807_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_808_Compl__iff,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
= ( ~ ( member_real @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_809_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_810_ComplD,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
=> ~ ( member_real @ C @ A2 ) ) ).
% ComplD
thf(fact_811_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A4: set_real] :
( collect_real
@ ^ [X2: real] :
~ ( member_real @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_812_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
~ ( member_nat @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_813_Compl__eq,axiom,
( uminus1532241313380277803et_int
= ( ^ [A4: set_int] :
( collect_int
@ ^ [X2: int] :
~ ( member_int @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_814_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_815_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ord_less_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A4 )
@ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_816_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_817_Collect__neg__eq,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
~ ( P @ X2 ) )
= ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% Collect_neg_eq
thf(fact_818_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A4: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_819_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_820_uminus__set__def,axiom,
( uminus1532241313380277803et_int
= ( ^ [A4: set_int] :
( collect_int
@ ( uminus_uminus_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_821_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_822_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A4 )
@ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_823_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_824_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_825_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_826_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_827_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_828_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A4 )
=> ( member_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_829_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_830_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A4 )
=> ( member_real @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_831_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_832_psubsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_833_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_834_subsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_835_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_836_in__mono,axiom,
! [A2: set_real,B3: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_837_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_838_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_839_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_840_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_841_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_842_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_843_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_844_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_845_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_846_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_847_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_848_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_849_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_850_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_851_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_852_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_853_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_854_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_855_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_856_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 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_857_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_858_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_859_less__account__def,axiom,
( ord_le2131251472502387783ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha_1 @ Alpha_2 )
& ~ ( ord_le4245800335709223507ccount @ Alpha_2 @ Alpha_1 ) ) ) ) ).
% less_account_def
thf(fact_860_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_861_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_862_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_863_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_864_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_865_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_866_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_867_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_868_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_869_pinf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_870_pinf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_871_pinf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_872_pinf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_873_pinf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_874_pinf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_875_pinf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_876_pinf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_877_pinf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_int @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_878_pinf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_879_pinf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_880_pinf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_int @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_881_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_882_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_883_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_884_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_885_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_886_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_887_minf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_888_minf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_889_minf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_890_minf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_891_minf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_892_minf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_893_minf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_894_minf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_895_minf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_int @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_896_minf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_897_minf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_898_minf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_int @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_899_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_900_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_901_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_902_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_903_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_904_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_905_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_906_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_907_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_908_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_909_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_910_infinite__descent0,axiom,
! [P: nat > $o,N2: 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 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_911_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_912_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_913_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_914_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_915_pinf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_916_pinf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_917_pinf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_918_pinf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_919_pinf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_920_pinf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_eq_int @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_921_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_922_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_923_Collect__empty__eq__bot,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( P = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_924_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_925_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X2: real] : ( member_real @ X2 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_926_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_927_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_928_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > real,B3: set_real] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_real @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_929_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > nat,B3: set_nat] :
( ! [X3: int] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( collect_int @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_930_image__Collect__subsetI,axiom,
! [P: int > $o,F: int > real,B3: set_real] :
( ! [X3: int] :
( ( P @ X3 )
=> ( member_real @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_int_real @ F @ ( collect_int @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_931_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_932_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_933_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_934_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ R )
@ ^ [X2: real] : ( member_real @ X2 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_935_prop__restrict,axiom,
! [X: real,Z5: set_real,X6: set_real,P: real > $o] :
( ( member_real @ X @ Z5 )
=> ( ( ord_less_eq_set_real @ Z5
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_936_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_937_prop__restrict,axiom,
! [X: int,Z5: set_int,X6: set_int,P: int > $o] :
( ( member_int @ X @ Z5 )
=> ( ( ord_less_eq_set_int @ Z5
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_938_Collect__restrict,axiom,
! [X6: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_939_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_940_Collect__restrict,axiom,
! [X6: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_941_subset__emptyI,axiom,
! [A2: set_real] :
( ! [X3: real] :
~ ( member_real @ X3 @ A2 )
=> ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_942_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_943_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_944_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_945_just__cash__valid__transfer,axiom,
! [C: real,T2: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T2 ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T2 )
& ( ord_less_eq_real @ T2 @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_946_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha2 ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_947_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_948_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_949_complete__interval,axiom,
! [A: real,B2: real,P: real > $o] :
( ( ord_less_real @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: real] :
( ( ord_less_eq_real @ A @ C4 )
& ( ord_less_eq_real @ C4 @ B2 )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_950_complete__interval,axiom,
! [A: int,B2: int,P: int > $o] :
( ( ord_less_int @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: int] :
( ( ord_less_eq_int @ A @ C4 )
& ( ord_less_eq_int @ C4 @ B2 )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_951_minus__Max__eq__Min,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( uminus_uminus_real @ ( lattic4275903605611617917x_real @ S ) )
= ( lattic3629708407755379051n_real @ ( image_real_real @ uminus_uminus_real @ S ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_952_minus__Max__eq__Min,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S ) )
= ( lattic8718645017227715691in_int @ ( image_int_int @ uminus_uminus_int @ S ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_953_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_954_Min_Obounded__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_955_Min_Obounded__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_956_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_957_Min__gr__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_real @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_958_Min__gr__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_959_Min__const,axiom,
! [A2: set_nat,C: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat
@ ( image_nat_nat
@ ^ [Uu: nat] : C
@ A2 ) )
= C ) ) ) ).
% Min_const
thf(fact_960_minus__Min__eq__Max,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( uminus_uminus_real @ ( lattic3629708407755379051n_real @ S ) )
= ( lattic4275903605611617917x_real @ ( image_real_real @ uminus_uminus_real @ S ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_961_minus__Min__eq__Max,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S ) )
= ( lattic8263393255366662781ax_int @ ( image_int_int @ uminus_uminus_int @ S ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_962_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_963_Min__le,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_964_Min__le,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_965_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_966_Min__eqI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ A2 )
=> ( ord_less_eq_real @ X @ Y2 ) )
=> ( ( member_real @ X @ A2 )
=> ( ( lattic3629708407755379051n_real @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_967_Min__eqI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A2 )
=> ( ord_less_eq_int @ X @ Y2 ) )
=> ( ( member_int @ X @ A2 )
=> ( ( lattic8718645017227715691in_int @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_968_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_969_Min_OcoboundedI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_970_Min_OcoboundedI,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_971_Min__in,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( member_real @ ( lattic3629708407755379051n_real @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_972_Min__in,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( member_int @ ( lattic8718645017227715691in_int @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_973_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_974_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_nat @ X @ A6 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_975_Min_OboundedI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [A6: real] :
( ( member_real @ A6 @ A2 )
=> ( ord_less_eq_real @ X @ A6 ) )
=> ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_976_Min_OboundedI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ord_less_eq_int @ X @ A6 ) )
=> ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_977_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_978_Min_OboundedE,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
=> ! [A8: real] :
( ( member_real @ A8 @ A2 )
=> ( ord_less_eq_real @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_979_Min_OboundedE,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
=> ! [A8: int] :
( ( member_int @ A8 @ A2 )
=> ( ord_less_eq_int @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_980_eq__Min__iff,axiom,
! [A2: set_nat,M3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M3
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat @ M3 @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M3 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_981_eq__Min__iff,axiom,
! [A2: set_real,M3: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( M3
= ( lattic3629708407755379051n_real @ A2 ) )
= ( ( member_real @ M3 @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ M3 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_982_eq__Min__iff,axiom,
! [A2: set_int,M3: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( M3
= ( lattic8718645017227715691in_int @ A2 ) )
= ( ( member_int @ M3 @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ M3 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_983_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_984_Min__le__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_985_Min__le__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_986_Min__eq__iff,axiom,
! [A2: set_nat,M3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M3 )
= ( ( member_nat @ M3 @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M3 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_987_Min__eq__iff,axiom,
! [A2: set_real,M3: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ( lattic3629708407755379051n_real @ A2 )
= M3 )
= ( ( member_real @ M3 @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ M3 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_988_Min__eq__iff,axiom,
! [A2: set_int,M3: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ( lattic8718645017227715691in_int @ A2 )
= M3 )
= ( ( member_int @ M3 @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ M3 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_989_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_990_Min__less__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic3629708407755379051n_real @ A2 ) @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_real @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_991_Min__less__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A2 ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_992_ex__gt__or__lt,axiom,
! [A: real] :
? [B6: real] :
( ( ord_less_real @ A @ B6 )
| ( ord_less_real @ B6 @ A ) ) ).
% ex_gt_or_lt
thf(fact_993_Min_Osubset__imp,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B3 ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_994_Min_Osubset__imp,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( finite_finite_real @ B3 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ B3 ) @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_995_Min_Osubset__imp,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( finite_finite_int @ B3 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B3 ) @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_996_Min__antimono,axiom,
! [M: set_nat,N3: set_nat] :
( ( ord_less_eq_set_nat @ M @ N3 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N3 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_997_Min__antimono,axiom,
! [M: set_real,N3: set_real] :
( ( ord_less_eq_set_real @ M @ N3 )
=> ( ( M != bot_bot_set_real )
=> ( ( finite_finite_real @ N3 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ N3 ) @ ( lattic3629708407755379051n_real @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_998_Min__antimono,axiom,
! [M: set_int,N3: set_int] :
( ( ord_less_eq_set_int @ M @ N3 )
=> ( ( M != bot_bot_set_int )
=> ( ( finite_finite_int @ N3 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N3 ) @ ( lattic8718645017227715691in_int @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_999_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_1000_dual__Max,axiom,
( ( lattices_Max_real
@ ^ [X2: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X2 ) )
= lattic3629708407755379051n_real ) ).
% dual_Max
thf(fact_1001_dual__Max,axiom,
( ( lattices_Max_int
@ ^ [X2: int,Y5: int] : ( ord_less_eq_int @ Y5 @ X2 ) )
= lattic8718645017227715691in_int ) ).
% dual_Max
thf(fact_1002_strictly__solvent__non__negative__cash,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_1003_bot_Oordering__top__axioms,axiom,
( ordering_top_set_nat
@ ^ [X2: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ X2 )
@ ^ [X2: set_nat,Y5: set_nat] : ( ord_less_set_nat @ Y5 @ X2 )
@ bot_bot_set_nat ) ).
% bot.ordering_top_axioms
thf(fact_1004_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_1005_arg__min__SOME__Min,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( lattic7446932960582359483at_nat @ F @ S )
= ( fChoice_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ S )
& ( ( F @ Y5 )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ S ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_1006_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1007_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_1008_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_1009_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_1010_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_1011_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_1012_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_1013_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_1014_strictly__solvent__just__cash__equiv,axiom,
! [C: real] :
( ( risk_F1636578016437888323olvent @ ( risk_Free_just_cash @ C ) )
= ( ord_less_eq_real @ zero_zero_real @ C ) ) ).
% strictly_solvent_just_cash_equiv
thf(fact_1015_some__in__eq,axiom,
! [A2: set_real] :
( ( member_real
@ ( fChoice_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_real ) ) ).
% some_in_eq
thf(fact_1016_some__in__eq,axiom,
! [A2: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_1017_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat] :
( ( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_1018_someI2__bex,axiom,
! [A2: set_real,P: real > $o,Q: real > $o] :
( ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: real] :
( ( ( member_real @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_1019_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > nat,X: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1020_subset__Collect__iff,axiom,
! [B3: set_real,A2: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B3 @ A2 )
=> ( ( ord_less_eq_set_real @ B3
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1021_subset__Collect__iff,axiom,
! [B3: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1022_subset__Collect__iff,axiom,
! [B3: set_int,A2: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B3 @ A2 )
=> ( ( ord_less_eq_set_int @ B3
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1023_subset__CollectI,axiom,
! [B3: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1024_subset__CollectI,axiom,
! [B3: set_int,A2: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B3 @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ B3 )
& ( Q @ X2 ) ) )
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1025_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1026_add__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1027_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1028_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1029_Rep__account__plus,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) )
= ( ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N ) ) ) ) ).
% Rep_account_plus
thf(fact_1030_just__cash__plus,axiom,
! [A: real,B2: real] :
( ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B2 ) )
= ( risk_Free_just_cash @ ( plus_plus_real @ A @ B2 ) ) ) ).
% just_cash_plus
thf(fact_1031_add__gr__0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1032_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1033_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1034_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1035_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1036_add__leE,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1037_le__add1,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) ) ).
% le_add1
thf(fact_1038_le__add2,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M3 @ N2 ) ) ).
% le_add2
thf(fact_1039_add__leD1,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% add_leD1
thf(fact_1040_add__leD2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1041_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1042_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1043_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1044_trans__le__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1045_trans__le__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1046_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1047_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1048_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1049_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1050_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1051_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1052_trans__less__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1053_trans__less__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1054_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1055_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1056_add__eq__self__zero,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= M3 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1057_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N ) ) ) ) ) ).
% plus_account_def
thf(fact_1058_additive__strictly__solvent,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_12 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_22 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_1059_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_less_as_int
thf(fact_1060_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_leq_as_int
thf(fact_1061_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1062_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1063_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1064_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1065_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1066_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1067_finite__interval__int1,axiom,
! [A: int,B2: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_eq_int @ I @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_1068_finite__interval__int4,axiom,
! [A: int,B2: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_int @ I @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_1069_negative__eq__positive,axiom,
! [N2: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N2 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1070_negative__zle,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zle
thf(fact_1071_finite__interval__int2,axiom,
! [A: int,B2: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_int @ I @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_1072_finite__interval__int3,axiom,
! [A: int,B2: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_eq_int @ I @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_1073_int__plus,axiom,
! [N2: nat,M3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% int_plus
thf(fact_1074_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_1075_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1076_int__ops_I5_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1077_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1078_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1079_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1080_not__int__zless__negative,axiom,
! [N2: nat,M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% not_int_zless_negative
thf(fact_1081_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1082_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1083_int__cases2,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_1084_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1085_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1086_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1087_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1088_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1089_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1090_int__zle__neg,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N2 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1091_zle__int,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% zle_int
thf(fact_1092_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1093_int__cases4,axiom,
! [M3: int] :
( ! [N4: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1094_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1095_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1096_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1097_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1098_nat__less__iff,axiom,
! [W3: int,M3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ M3 )
= ( ord_less_int @ W3 @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ).
% nat_less_iff
thf(fact_1099_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1100_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1101_zless__nat__conj,axiom,
! [W3: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W3 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1102_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1103_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1104_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1105_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1106_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1107_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1108_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1109_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1110_nat__mono__iff,axiom,
! [Z: int,W3: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W3 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1111_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1112_zless__nat__eq__int__zless,axiom,
! [M3: nat,Z: int] :
( ( ord_less_nat @ M3 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1113_int__eq__iff,axiom,
! [M3: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M3 )
= Z )
= ( ( M3
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1114_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1115_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1116_nat__less__eq__zless,axiom,
! [W3: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( ( ord_less_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W3 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1117_nat__le__eq__zle,axiom,
! [W3: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W3 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W3 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W3 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1118_nat__eq__iff2,axiom,
! [M3: nat,W3: int] :
( ( M3
= ( nat2 @ W3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1119_nat__eq__iff,axiom,
! [W3: int,M3: nat] :
( ( ( nat2 @ W3 )
= M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( W3
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W3 )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1120_split__nat,axiom,
! [P: nat > $o,I2: int] :
( ( P @ ( nat2 @ I2 ) )
= ( ! [N: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N ) )
=> ( P @ N ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1121_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1122_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1123_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_1124_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N: nat] :
( ( N != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_1125_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1126_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1127_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1128_zle__add1__eq__le,axiom,
! [W3: int,Z: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1129_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1130_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1131_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1132_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1133_zless__add1__eq,axiom,
! [W3: int,Z: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z )
| ( W3 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1134_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1135_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1136_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1137_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_1138_zless__imp__add1__zle,axiom,
! [W3: int,Z: int] :
( ( ord_less_int @ W3 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1139_add1__zle__eq,axiom,
! [W3: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z )
= ( ord_less_int @ W3 @ Z ) ) ).
% add1_zle_eq
thf(fact_1140_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1141_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1142_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% ln_inverse
thf(fact_1160_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_1161_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1162_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1163_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1164_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1165_Suc__mono,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1166_Suc__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1167_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1168_add__Suc__right,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1169_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1170_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1171_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1172_negative__zless,axiom,
! [N2: nat,M3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zless
thf(fact_1173_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1174_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1175_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ M3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1176_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1177_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1178_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1179_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1180_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1181_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1182_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1183_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1184_not__less__less__Suc__eq,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1185_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1186_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1187_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1188_Suc__less__SucD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1189_less__antisym,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
=> ( M3 = N2 ) ) ) ).
% less_antisym
thf(fact_1190_Suc__less__eq2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1191_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1192_not__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M3 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1193_less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1194_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1195_less__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1196_less__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M3 @ N2 )
=> ( M3 = N2 ) ) ) ).
% less_SucE
thf(fact_1197_Suc__lessI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ( suc @ M3 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1198_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1199_Suc__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_lessD
thf(fact_1200_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1201_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1202_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1203_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1204_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1205_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1206_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1207_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1208_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1209_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1210_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1211_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1212_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1213_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1214_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1215_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1216_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1217_add__Suc,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc
thf(fact_1218_add__Suc__shift,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( plus_plus_nat @ M3 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1219_nat__power__eq,axiom,
! [Z: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_1220_int__cases,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% int_cases
thf(fact_1221_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
=> ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1222_less__imp__Suc__add,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1223_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1224_less__add__Suc2,axiom,
! [I2: nat,M3: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M3 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1225_less__add__Suc1,axiom,
! [I2: nat,M3: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1226_less__natE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M3 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1227_one__is__add,axiom,
! [M3: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1228_add__is__1,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1229_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1230_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1231_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1232_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1233_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1234_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1235_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1236_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1237_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1238_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1239_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1240_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1241_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1242_less__Suc__eq__0__disj,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( M3 = zero_zero_nat )
| ? [J3: nat] :
( ( M3
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1243_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_1244_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1245_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1246_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1247_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1248_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1249_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_1250_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1251_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1252_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1253_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1254_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_1255_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N4: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% negD
thf(fact_1256_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1257_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1258_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1259_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1260_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1261_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M3: nat] :
( ( ( power_power_nat @ X @ M3 )
= ( suc @ zero_zero_nat ) )
= ( ( M3 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1262_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_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 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X8: nat] : ( P @ X8 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: real > $o] :
( ( P @ ( fChoice_real @ P ) )
= ( ? [X8: real] : ( P @ X8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( risk_F170160801229183585ccount @ alpha @ ( risk_F4612863212915232279period @ alpha ) )
!= zero_zero_real ) ).
%------------------------------------------------------------------------------