TPTP Problem File: SLH0258^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Risk_Free_Lending/0000_Risk_Free_Lending/prob_01092_034328__5946796_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1591 ( 547 unt; 318 typ; 0 def)
% Number of atoms : 3881 (1119 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 9392 ( 383 ~; 77 |; 202 &;7087 @)
% ( 0 <=>;1643 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 67 ( 66 usr)
% Number of type conns : 879 ( 879 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 252 usr; 52 con; 0-3 aty)
% Number of variables : 3190 ( 208 ^;2813 !; 169 ?;3190 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:32.722
%------------------------------------------------------------------------------
% Could-be-implicit typings (66)
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
top_to4146629662433032470on_nat: set_list_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
top_to2669572386385008721t_real: set_option_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
top_to2947398533425636438on_nat: set_op6961666426309957030on_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
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top_to6565372977178185615at_nat: set_Pr1076074158304066111at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
top_to4387612366039908569ccount: set_Ri1641125681238393385ccount ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
top_to1863808837862421559t_real: set_set_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
top_to3692820865130894140on_nat: set_set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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top_to497527773527737537t_real: set_Sum_sum_nat_real ).
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top_to6095207040221017013at_nat: set_Su1228339496316593413at_nat ).
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top_to3687521912721340737al_nat: set_Sum_sum_real_nat ).
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top_to8895904057622651549l_real: set_Su3647026645378120685l_real ).
thf(sy_c_Ordinal__Arithmetic_Ofin__support_001t__Real__Oreal_001t__Nat__Onat,type,
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ordina7525502726642723294al_nat: real > set_nat > ( nat > real ) > set_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Onet__asset__value,type,
risk_F2906766666041932210_value: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_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__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Nat__Onat_J,type,
collect_option_nat: ( option_nat > $o ) > set_option_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec1753597841410690900at_nat: ( sum_sum_nat_nat > $o ) > set_Sum_sum_nat_nat ).
thf(sy_c_Set_OPow_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
pow_nat_real: set_nat_real > set_set_nat_real ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Option__Ooption_It__Nat__Onat_J,type,
pow_option_nat: set_option_nat > set_set_option_nat ).
thf(sy_c_Set_OPow_001t__Real__Oreal,type,
pow_real: set_real > set_set_real ).
thf(sy_c_Set_OPow_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
pow_Sum_sum_nat_nat: set_Sum_sum_nat_nat > set_se3873067930692246379at_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
sum_Pl2515177958263135456t_real: set_nat > set_nat_real > set_Su413319825862275584t_real ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Plus_nat_nat: set_nat > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
sum_Pl7347870528396357733on_nat: set_nat > set_option_nat > set_Su4129973685039490437on_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Real__Oreal,type,
sum_Plus_nat_real: set_nat > set_real > set_Sum_sum_nat_real ).
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thf(sy_c_Sum__Type_OPlus_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
sum_Pl2045061157817691621at_nat: set_option_nat > set_nat > set_Su1228339496316593413at_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
sum_Pl2720954102514207669on_nat: set_option_nat > set_option_nat > set_Su5175481047276092373on_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Option__Ooption_It__Nat__Onat_J_001t__Real__Oreal,type,
sum_Pl3732505016401745473t_real: set_option_nat > set_real > set_Su6341430339211832417t_real ).
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thf(sy_c_Sum__Type_OPlus_001t__Real__Oreal_001t__Nat__Onat,type,
sum_Plus_real_nat: set_real > set_nat > set_Sum_sum_real_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Real__Oreal_001t__Option__Ooption_It__Nat__Onat_J,type,
sum_Pl192440012131237697on_nat: set_real > set_option_nat > set_Su8421005785618960225on_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Real__Oreal_001t__Real__Oreal,type,
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type_d3639048291015846244t_real: ( ( nat > real ) > nat > real ) > ( ( nat > real ) > nat > real ) > set_nat_real > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Typedef_Otype__definition_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
type_d8392851985301059310on_nat: ( option_nat > option_nat ) > ( option_nat > option_nat ) > set_option_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
member_nat_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_member_001t__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__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_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,
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thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
member_set_nat_real: set_nat_real > set_set_nat_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
member3860231779568403053on_nat: set_option_nat > set_set_option_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_v__092_060alpha_062,type,
alpha: risk_Free_account ).
% Relevant facts (1267)
thf(fact_0_Rep__account,axiom,
! [X: risk_Free_account] : ( member_nat_real @ ( risk_F170160801229183585ccount @ X ) @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ).
% Rep_account
thf(fact_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_Rep__account__cases,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ~ ! [X2: risk_Free_account] :
( Y
!= ( risk_F170160801229183585ccount @ X2 ) ) ) ).
% Rep_account_cases
thf(fact_3_Rep__account__induct,axiom,
! [Y: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ! [X2: risk_Free_account] : ( P @ ( risk_F170160801229183585ccount @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_account_induct
thf(fact_4_finite__option__UNIV,axiom,
( ( finite4243986238680484743on_nat @ top_to2947398533425636438on_nat )
= ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ).
% finite_option_UNIV
thf(fact_5_finite__option__UNIV,axiom,
( ( finite8881611996251951590at_nat @ top_to7746642472848562997at_nat )
= ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat ) ) ).
% finite_option_UNIV
thf(fact_6_finite__option__UNIV,axiom,
( ( finite5731380839103853331n_real @ top_to853713521313446370n_real )
= ( finite_finite_real @ top_top_set_real ) ) ).
% finite_option_UNIV
thf(fact_7_finite__option__UNIV,axiom,
( ( finite4829245213595033346t_real @ top_to2669572386385008721t_real )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ).
% finite_option_UNIV
thf(fact_8_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_9_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_10_finite__Plus__UNIV__iff,axiom,
( ( finite4392130603907730802t_real @ top_to497527773527737537t_real )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_11_finite__Plus__UNIV__iff,axiom,
( ( finite417082912430850546al_nat @ top_to3687521912721340737al_nat )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_12_finite__Plus__UNIV__iff,axiom,
( ( finite739467132598283086l_real @ top_to8895904057622651549l_real )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_13_finite__Plus__UNIV__iff,axiom,
( ( finite4954261698061101798on_nat @ top_to8996841228943914037on_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_14_finite__Plus__UNIV__iff,axiom,
( ( finite2615695030181177958at_nat @ top_to6095207040221017013at_nat )
= ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_15_finite__Plus__UNIV__iff,axiom,
( ( finite1324221693455276482t_real @ top_to5034386609897464593t_real )
= ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_16_finite__Plus__UNIV__iff,axiom,
( ( finite4724301795589799106on_nat @ top_to7113962056304592401on_nat )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_17_finite__Plus__UNIV__iff,axiom,
( ( finite1365476281568474437at_nat @ top_to2078852577765846804at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_18_finite__Plus__UNIV__iff,axiom,
( ( finite675431524867096289t_real @ top_to8329648133402683568t_real )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_19_UNIV__I,axiom,
! [X: option_nat] : ( member_option_nat @ X @ top_to8920198386146353926on_nat ) ).
% UNIV_I
thf(fact_20_UNIV__I,axiom,
! [X: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X @ top_to6661820994512907621at_nat ) ).
% UNIV_I
thf(fact_21_UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% UNIV_I
thf(fact_22_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_23_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_24_iso__tuple__UNIV__I,axiom,
! [X: option_nat] : ( member_option_nat @ X @ top_to8920198386146353926on_nat ) ).
% iso_tuple_UNIV_I
thf(fact_25_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X @ top_to6661820994512907621at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_26_iso__tuple__UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% iso_tuple_UNIV_I
thf(fact_27_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_28_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_29_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_30_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_31_Finite__Set_Ofinite__set,axiom,
( ( finite1464753433994532717on_nat @ top_to3692820865130894140on_nat )
= ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ).
% Finite_Set.finite_set
thf(fact_32_Finite__Set_Ofinite__set,axiom,
( ( finite5325900196762371532at_nat @ top_to6951743751474147867at_nat )
= ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat ) ) ).
% Finite_Set.finite_set
thf(fact_33_Finite__Set_Ofinite__set,axiom,
( ( finite9007344921179782393t_real @ top_top_set_set_real )
= ( finite_finite_real @ top_top_set_real ) ) ).
% Finite_Set.finite_set
thf(fact_34_Finite__Set_Ofinite__set,axiom,
( ( finite7096078154069415912t_real @ top_to1863808837862421559t_real )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ).
% Finite_Set.finite_set
thf(fact_35_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_36_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_37_top__set__def,axiom,
( top_to8920198386146353926on_nat
= ( collect_option_nat @ top_top_option_nat_o ) ) ).
% top_set_def
thf(fact_38_top__set__def,axiom,
( top_to6661820994512907621at_nat
= ( collec1753597841410690900at_nat @ top_to8066396485052406624_nat_o ) ) ).
% top_set_def
thf(fact_39_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_40_top__set__def,axiom,
( top_top_set_nat_real
= ( collect_nat_real @ top_top_nat_real_o ) ) ).
% top_set_def
thf(fact_41_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_42_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_43_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_44_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_45_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_46_UNIV__witness,axiom,
? [X2: option_nat] : ( member_option_nat @ X2 @ top_to8920198386146353926on_nat ) ).
% UNIV_witness
thf(fact_47_UNIV__witness,axiom,
? [X2: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X2 @ top_to6661820994512907621at_nat ) ).
% UNIV_witness
thf(fact_48_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_49_UNIV__witness,axiom,
? [X2: nat > real] : ( member_nat_real @ X2 @ top_top_set_nat_real ) ).
% UNIV_witness
thf(fact_50_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_51_UNIV__eq__I,axiom,
! [A: set_option_nat] :
( ! [X2: option_nat] : ( member_option_nat @ X2 @ A )
=> ( top_to8920198386146353926on_nat = A ) ) ).
% UNIV_eq_I
thf(fact_52_UNIV__eq__I,axiom,
! [A: set_Sum_sum_nat_nat] :
( ! [X2: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X2 @ A )
=> ( top_to6661820994512907621at_nat = A ) ) ).
% UNIV_eq_I
thf(fact_53_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_54_UNIV__eq__I,axiom,
! [A: set_nat_real] :
( ! [X2: nat > real] : ( member_nat_real @ X2 @ A )
=> ( top_top_set_nat_real = A ) ) ).
% UNIV_eq_I
thf(fact_55_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_56_infinite__UNIV__char__0,axiom,
~ ( finite_finite_real @ top_top_set_real ) ).
% infinite_UNIV_char_0
thf(fact_57_finite__fun__UNIVD2,axiom,
( ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ( finite_finite_real @ top_top_set_real ) ) ).
% finite_fun_UNIVD2
thf(fact_58_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_59_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_real @ top_top_set_real )
=> ( finite7458326234658656262t_real @ top_to4855536200657754381t_real ) ) ) ).
% finite_Prod_UNIV
thf(fact_60_finite__Prod__UNIV,axiom,
( ( finite_finite_real @ top_top_set_real )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite3483278543181776006al_nat @ top_to8045530339851357581al_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_61_finite__Prod__UNIV,axiom,
( ( finite_finite_real @ top_top_set_real )
=> ( ( finite_finite_real @ top_top_set_real )
=> ( finite4592917586124760034l_real @ top_to1175844992842649833l_real ) ) ) ).
% finite_Prod_UNIV
thf(fact_62_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ( finite8991716528752003450on_nat @ top_to3158204655393688705on_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_63_finite__Prod__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6653149860872079610at_nat @ top_to256570466670791681at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_64_finite__Prod__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ( ( finite_finite_real @ top_top_set_real )
=> ( finite4980775327857552982t_real @ top_to830787761519449437t_real ) ) ) ).
% finite_Prod_UNIV
thf(fact_65_finite__Prod__UNIV,axiom,
( ( finite_finite_real @ top_top_set_real )
=> ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ( finite8380855429992075606on_nat @ top_to2910363207926577245on_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_66_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
=> ( finite6010018676071990233at_nat @ top_to678699698364323680at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_67_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ( finite4331985159269372789t_real @ top_to4126049285024668412t_real ) ) ) ).
% finite_Prod_UNIV
thf(fact_68_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A2: nat] :
~ ( member_nat @ A2 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_69_ex__new__if__finite,axiom,
! [A: set_option_nat] :
( ~ ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ( ( finite5523153139673422903on_nat @ A )
=> ? [A2: option_nat] :
~ ( member_option_nat @ A2 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_70_ex__new__if__finite,axiom,
! [A: set_Sum_sum_nat_nat] :
( ~ ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
=> ( ( finite6187706683773761046at_nat @ A )
=> ? [A2: sum_sum_nat_nat] :
~ ( member8583185029347631382at_nat @ A2 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_71_ex__new__if__finite,axiom,
! [A: set_real] :
( ~ ( finite_finite_real @ top_top_set_real )
=> ( ( finite_finite_real @ A )
=> ? [A2: real] :
~ ( member_real @ A2 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_72_ex__new__if__finite,axiom,
! [A: set_nat_real] :
( ~ ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ( ( finite7853608736407863218t_real @ A )
=> ? [A2: nat > real] :
~ ( member_nat_real @ A2 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_73_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_74_finite__prod,axiom,
( ( finite7458326234658656262t_real @ top_to4855536200657754381t_real )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_prod
thf(fact_75_finite__prod,axiom,
( ( finite3483278543181776006al_nat @ top_to8045530339851357581al_nat )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_76_finite__prod,axiom,
( ( finite4592917586124760034l_real @ top_to1175844992842649833l_real )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_prod
thf(fact_77_finite__prod,axiom,
( ( finite8991716528752003450on_nat @ top_to3158204655393688705on_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ) ).
% finite_prod
thf(fact_78_finite__prod,axiom,
( ( finite6653149860872079610at_nat @ top_to256570466670791681at_nat )
= ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_79_finite__prod,axiom,
( ( finite4980775327857552982t_real @ top_to830787761519449437t_real )
= ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_prod
thf(fact_80_finite__prod,axiom,
( ( finite8380855429992075606on_nat @ top_to2910363207926577245on_nat )
= ( ( finite_finite_real @ top_top_set_real )
& ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ) ).
% finite_prod
thf(fact_81_finite__prod,axiom,
( ( finite6010018676071990233at_nat @ top_to678699698364323680at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat ) ) ) ).
% finite_prod
thf(fact_82_finite__prod,axiom,
( ( finite4331985159269372789t_real @ top_to4126049285024668412t_real )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ) ).
% finite_prod
thf(fact_83_finite__support,axiom,
! [F: nat > real,Z: real,A: set_nat] :
( ( member_nat_real @ F @ ( ordina1579063754167848977al_nat @ Z @ A ) )
=> ( finite_finite_nat @ ( ordina7525502726642723294al_nat @ Z @ A @ F ) ) ) ).
% finite_support
thf(fact_84_Abs__account__inverse,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( risk_F170160801229183585ccount @ ( risk_F5458100604530014700ccount @ Y ) )
= Y ) ) ).
% Abs_account_inverse
thf(fact_85_Abs__account__inject,axiom,
! [X: nat > real,Y: nat > real] :
( ( member_nat_real @ X @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( ( risk_F5458100604530014700ccount @ X )
= ( risk_F5458100604530014700ccount @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_account_inject
thf(fact_86_Abs__account__induct,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( P @ ( risk_F5458100604530014700ccount @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_account_induct
thf(fact_87_Abs__account__cases,axiom,
! [X: risk_Free_account] :
~ ! [Y2: nat > real] :
( ( X
= ( risk_F5458100604530014700ccount @ Y2 ) )
=> ~ ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ) ).
% Abs_account_cases
thf(fact_88_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_89_shortest__period___092_060pi_062,axiom,
! [Alpha2: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha2 @ I )
!= zero_zero_real )
=> ( ( risk_F170160801229183585ccount @ Alpha2 @ ( risk_F4612863212915232279period @ Alpha2 ) )
!= zero_zero_real ) ) ).
% shortest_period_\<pi>
thf(fact_90_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_91_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_92_top__empty__eq,axiom,
( top_top_option_nat_o
= ( ^ [X3: option_nat] : ( member_option_nat @ X3 @ top_to8920198386146353926on_nat ) ) ) ).
% top_empty_eq
thf(fact_93_top__empty__eq,axiom,
( top_to8066396485052406624_nat_o
= ( ^ [X3: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X3 @ top_to6661820994512907621at_nat ) ) ) ).
% top_empty_eq
thf(fact_94_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_95_top__empty__eq,axiom,
( top_top_nat_real_o
= ( ^ [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ) ) ).
% top_empty_eq
thf(fact_96_finite__Plus__iff,axiom,
! [A: set_nat,B: set_option_nat] :
( ( finite4954261698061101798on_nat @ ( sum_Pl7347870528396357733on_nat @ A @ B ) )
= ( ( finite_finite_nat @ A )
& ( finite5523153139673422903on_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_97_finite__Plus__iff,axiom,
! [A: set_nat,B: set_Sum_sum_nat_nat] :
( ( finite1365476281568474437at_nat @ ( sum_Pl7753636632751542852at_nat @ A @ B ) )
= ( ( finite_finite_nat @ A )
& ( finite6187706683773761046at_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_98_finite__Plus__iff,axiom,
! [A: set_option_nat,B: set_nat] :
( ( finite2615695030181177958at_nat @ ( sum_Pl2045061157817691621at_nat @ A @ B ) )
= ( ( finite5523153139673422903on_nat @ A )
& ( finite_finite_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_99_finite__Plus__iff,axiom,
! [A: set_option_nat,B: set_option_nat] :
( ( finite5059127062945834806on_nat @ ( sum_Pl2720954102514207669on_nat @ A @ B ) )
= ( ( finite5523153139673422903on_nat @ A )
& ( finite5523153139673422903on_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_100_finite__Plus__iff,axiom,
! [A: set_option_nat,B: set_Sum_sum_nat_nat] :
( ( finite8672868503146279317at_nat @ ( sum_Pl7266259019762183572at_nat @ A @ B ) )
= ( ( finite5523153139673422903on_nat @ A )
& ( finite6187706683773761046at_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_101_finite__Plus__iff,axiom,
! [A: set_Sum_sum_nat_nat,B: set_nat] :
( ( finite7838611791898061893at_nat @ ( sum_Pl8395311126014171076at_nat @ A @ B ) )
= ( ( finite6187706683773761046at_nat @ A )
& ( finite_finite_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_102_finite__Plus__iff,axiom,
! [A: set_Sum_sum_nat_nat,B: set_option_nat] :
( ( finite2086700240624447125on_nat @ ( sum_Pl7036244479726910228on_nat @ A @ B ) )
= ( ( finite6187706683773761046at_nat @ A )
& ( finite5523153139673422903on_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_103_finite__Plus__iff,axiom,
! [A: set_Sum_sum_nat_nat,B: set_Sum_sum_nat_nat] :
( ( finite5180632770264759540at_nat @ ( sum_Pl4627831089307508979at_nat @ A @ B ) )
= ( ( finite6187706683773761046at_nat @ A )
& ( finite6187706683773761046at_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_104_finite__Plus__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) )
= ( ( finite_finite_nat @ A )
& ( finite_finite_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_105_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_106_Pow__UNIV,axiom,
( ( pow_option_nat @ top_to8920198386146353926on_nat )
= top_to3692820865130894140on_nat ) ).
% Pow_UNIV
thf(fact_107_Pow__UNIV,axiom,
( ( pow_Sum_sum_nat_nat @ top_to6661820994512907621at_nat )
= top_to6951743751474147867at_nat ) ).
% Pow_UNIV
thf(fact_108_Pow__UNIV,axiom,
( ( pow_real @ top_top_set_real )
= top_top_set_set_real ) ).
% Pow_UNIV
thf(fact_109_Pow__UNIV,axiom,
( ( pow_nat_real @ top_top_set_nat_real )
= top_to1863808837862421559t_real ) ).
% Pow_UNIV
thf(fact_110_type__definition__account,axiom,
type_d8982087200295354172t_real @ risk_F170160801229183585ccount @ risk_F5458100604530014700ccount @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ).
% type_definition_account
thf(fact_111_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_112_finite__Pow__iff,axiom,
! [A: set_option_nat] :
( ( finite1464753433994532717on_nat @ ( pow_option_nat @ A ) )
= ( finite5523153139673422903on_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_113_finite__Pow__iff,axiom,
! [A: set_Sum_sum_nat_nat] :
( ( finite5325900196762371532at_nat @ ( pow_Sum_sum_nat_nat @ A ) )
= ( finite6187706683773761046at_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_114_just__cash__embed,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( risk_Free_just_cash @ A3 )
= ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_embed
thf(fact_115_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_116_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_117_finite__PlusD_I2_J,axiom,
! [A: set_nat,B: set_nat] :
( ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) )
=> ( finite_finite_nat @ B ) ) ).
% finite_PlusD(2)
thf(fact_118_mem__Collect__eq,axiom,
! [A4: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A4 @ ( collect_nat_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
! [A4: real,P: real > $o] :
( ( member_real @ A4 @ ( collect_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_nat_real] :
( ( collect_nat_real
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_124_finite__PlusD_I1_J,axiom,
! [A: set_nat,B: set_nat] :
( ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) )
=> ( finite_finite_nat @ A ) ) ).
% finite_PlusD(1)
thf(fact_125_finite__Plus,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_126_finite__Plus,axiom,
! [A: set_nat,B: set_option_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite5523153139673422903on_nat @ B )
=> ( finite4954261698061101798on_nat @ ( sum_Pl7347870528396357733on_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_127_finite__Plus,axiom,
! [A: set_nat,B: set_Sum_sum_nat_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite6187706683773761046at_nat @ B )
=> ( finite1365476281568474437at_nat @ ( sum_Pl7753636632751542852at_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_128_finite__Plus,axiom,
! [A: set_option_nat,B: set_nat] :
( ( finite5523153139673422903on_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2615695030181177958at_nat @ ( sum_Pl2045061157817691621at_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_129_finite__Plus,axiom,
! [A: set_option_nat,B: set_option_nat] :
( ( finite5523153139673422903on_nat @ A )
=> ( ( finite5523153139673422903on_nat @ B )
=> ( finite5059127062945834806on_nat @ ( sum_Pl2720954102514207669on_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_130_finite__Plus,axiom,
! [A: set_option_nat,B: set_Sum_sum_nat_nat] :
( ( finite5523153139673422903on_nat @ A )
=> ( ( finite6187706683773761046at_nat @ B )
=> ( finite8672868503146279317at_nat @ ( sum_Pl7266259019762183572at_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_131_finite__Plus,axiom,
! [A: set_Sum_sum_nat_nat,B: set_nat] :
( ( finite6187706683773761046at_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite7838611791898061893at_nat @ ( sum_Pl8395311126014171076at_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_132_finite__Plus,axiom,
! [A: set_Sum_sum_nat_nat,B: set_option_nat] :
( ( finite6187706683773761046at_nat @ A )
=> ( ( finite5523153139673422903on_nat @ B )
=> ( finite2086700240624447125on_nat @ ( sum_Pl7036244479726910228on_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_133_finite__Plus,axiom,
! [A: set_Sum_sum_nat_nat,B: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ A )
=> ( ( finite6187706683773761046at_nat @ B )
=> ( finite5180632770264759540at_nat @ ( sum_Pl4627831089307508979at_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_134_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_nat_nat @ top_top_set_nat @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_135_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_nat_real @ top_top_set_nat @ top_top_set_real )
= top_to497527773527737537t_real ) ).
% UNIV_Plus_UNIV
thf(fact_136_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_real_nat @ top_top_set_real @ top_top_set_nat )
= top_to3687521912721340737al_nat ) ).
% UNIV_Plus_UNIV
thf(fact_137_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_real_real @ top_top_set_real @ top_top_set_real )
= top_to8895904057622651549l_real ) ).
% UNIV_Plus_UNIV
thf(fact_138_UNIV__Plus__UNIV,axiom,
( ( sum_Pl7347870528396357733on_nat @ top_top_set_nat @ top_to8920198386146353926on_nat )
= top_to8996841228943914037on_nat ) ).
% UNIV_Plus_UNIV
thf(fact_139_UNIV__Plus__UNIV,axiom,
( ( sum_Pl2045061157817691621at_nat @ top_to8920198386146353926on_nat @ top_top_set_nat )
= top_to6095207040221017013at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_140_UNIV__Plus__UNIV,axiom,
( ( sum_Pl3732505016401745473t_real @ top_to8920198386146353926on_nat @ top_top_set_real )
= top_to5034386609897464593t_real ) ).
% UNIV_Plus_UNIV
thf(fact_141_UNIV__Plus__UNIV,axiom,
( ( sum_Pl192440012131237697on_nat @ top_top_set_real @ top_to8920198386146353926on_nat )
= top_to7113962056304592401on_nat ) ).
% UNIV_Plus_UNIV
thf(fact_142_UNIV__Plus__UNIV,axiom,
( ( sum_Pl7753636632751542852at_nat @ top_top_set_nat @ top_to6661820994512907621at_nat )
= top_to2078852577765846804at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_143_UNIV__Plus__UNIV,axiom,
( ( sum_Pl2515177958263135456t_real @ top_top_set_nat @ top_top_set_nat_real )
= top_to8329648133402683568t_real ) ).
% UNIV_Plus_UNIV
thf(fact_144_infinite__Pow,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ~ ( finite1152437895449049373et_nat @ ( pow_nat @ A ) ) ) ).
% infinite_Pow
thf(fact_145_infinite__Pow,axiom,
! [A: set_option_nat] :
( ~ ( finite5523153139673422903on_nat @ A )
=> ~ ( finite1464753433994532717on_nat @ ( pow_option_nat @ A ) ) ) ).
% infinite_Pow
thf(fact_146_infinite__Pow,axiom,
! [A: set_Sum_sum_nat_nat] :
( ~ ( finite6187706683773761046at_nat @ A )
=> ~ ( finite5325900196762371532at_nat @ ( pow_Sum_sum_nat_nat @ A ) ) ) ).
% infinite_Pow
thf(fact_147_type__copy__ex__RepI,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,F2: ( nat > real ) > $o] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ( ( ? [X4: nat > real] : ( F2 @ X4 ) )
= ( ? [B2: risk_Free_account] : ( F2 @ ( Rep @ B2 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_148_type__copy__obj__one__point__absE,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,S: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ~ ! [X2: nat > real] :
( S
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_149_net__asset__value__zero,axiom,
( ( risk_F2906766666041932210_value @ zero_z1425366712893667068ccount )
= zero_zero_real ) ).
% net_asset_value_zero
thf(fact_150_type__definition__def,axiom,
( type_d8982087200295354172t_real
= ( ^ [Rep2: risk_Free_account > nat > real,Abs2: ( nat > real ) > risk_Free_account,A5: set_nat_real] :
( ! [X3: risk_Free_account] : ( member_nat_real @ ( Rep2 @ X3 ) @ A5 )
& ! [X3: risk_Free_account] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y4: nat > real] :
( ( member_nat_real @ Y4 @ A5 )
=> ( ( Rep2 @ ( Abs2 @ Y4 ) )
= Y4 ) ) ) ) ) ).
% type_definition_def
thf(fact_151_type__definition_ORep__inverse,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,X: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_152_type__definition_OAbs__inverse,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,Y: nat > real] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( member_nat_real @ Y @ A )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_153_type__definition_ORep__inject,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,X: risk_Free_account,Y: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_154_net__asset__value__just__cash__left__inverse,axiom,
! [C: real] :
( ( risk_F2906766666041932210_value @ ( risk_Free_just_cash @ C ) )
= C ) ).
% net_asset_value_just_cash_left_inverse
thf(fact_155_type__definition_ORep,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,X: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( member_nat_real @ ( Rep @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_156_type__definition_Ointro,axiom,
! [Rep: risk_Free_account > nat > real,A: set_nat_real,Abs: ( nat > real ) > risk_Free_account] :
( ! [X2: risk_Free_account] : ( member_nat_real @ ( Rep @ X2 ) @ A )
=> ( ! [X2: risk_Free_account] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ A )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d8982087200295354172t_real @ Rep @ Abs @ A ) ) ) ) ).
% type_definition.intro
thf(fact_157_type__definition_OAbs__cases,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,X: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ~ ! [Y2: nat > real] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member_nat_real @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_158_type__definition_ORep__cases,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,Y: nat > real] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( member_nat_real @ Y @ A )
=> ~ ! [X2: risk_Free_account] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_159_type__definition_OAbs__induct,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,P: risk_Free_account > $o,X: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ A )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_160_type__definition_OAbs__inject,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,X: nat > real,Y: nat > real] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( member_nat_real @ X @ A )
=> ( ( member_nat_real @ Y @ A )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_161_type__definition_ORep__induct,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,A: set_nat_real,Y: nat > real,P: ( nat > real ) > $o] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ A )
=> ( ( member_nat_real @ Y @ A )
=> ( ! [X2: risk_Free_account] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_162_shortest__period__bound,axiom,
! [Alpha2: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha2 @ I )
!= zero_zero_real )
=> ( ord_less_eq_nat @ I @ ( risk_F4612863212915232279period @ Alpha2 ) ) ) ).
% shortest_period_bound
thf(fact_163_greater__than__shortest__period__zero,axiom,
! [Alpha2: risk_Free_account,M: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ Alpha2 ) @ M )
=> ( ( risk_F170160801229183585ccount @ Alpha2 @ M )
= zero_zero_real ) ) ).
% greater_than_shortest_period_zero
thf(fact_164_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d6250493948777748686at_nat @ bNF_id_bnf_nat @ bNF_id_bnf_nat @ top_top_set_nat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_165_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d8392851985301059310on_nat @ bNF_id7546324870103583748on_nat @ bNF_id7546324870103583748on_nat @ top_to8920198386146353926on_nat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_166_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d2961426133912454188at_nat @ bNF_id970554881599238627at_nat @ bNF_id970554881599238627at_nat @ top_to6661820994512907621at_nat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_167_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d2211904104766253830l_real @ bNF_id_bnf_real @ bNF_id_bnf_real @ top_top_set_real ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_168_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d3639048291015846244t_real @ bNF_id_bnf_nat_real @ bNF_id_bnf_nat_real @ top_top_set_nat_real ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_169_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_170_Times__same__infinite__bij__betw__types,axiom,
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ? [F3: product_prod_nat_nat > nat] : ( bij_be5333170631980326235at_nat @ F3 @ top_to4669805908274784177at_nat @ top_top_set_nat ) ) ).
% Times_same_infinite_bij_betw_types
thf(fact_171_Times__same__infinite__bij__betw__types,axiom,
( ~ ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ? [F3: produc4953844613479565601on_nat > option_nat] : ( bij_be7152475446971823947on_nat @ F3 @ top_to6363332153347717329on_nat @ top_to8920198386146353926on_nat ) ) ).
% Times_same_infinite_bij_betw_types
thf(fact_172_Times__same__infinite__bij__betw__types,axiom,
( ~ ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
=> ? [F3: produc3819362883047884511at_nat > sum_sum_nat_nat] : ( bij_be8160692440542302696at_nat @ F3 @ top_to6565372977178185615at_nat @ top_to6661820994512907621at_nat ) ) ).
% Times_same_infinite_bij_betw_types
thf(fact_173_Times__same__infinite__bij__betw__types,axiom,
( ~ ( finite_finite_real @ top_top_set_real )
=> ? [F3: produc2422161461964618553l_real > real] : ( bij_be2994174799932477423l_real @ F3 @ top_to1175844992842649833l_real @ top_top_set_real ) ) ).
% Times_same_infinite_bij_betw_types
thf(fact_174_Times__same__infinite__bij__betw__types,axiom,
( ~ ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ? [F3: produc6204762708310045719t_real > nat > real] : ( bij_be1765362612115117756t_real @ F3 @ top_to6916931193077932999t_real @ top_top_set_nat_real ) ) ).
% Times_same_infinite_bij_betw_types
thf(fact_175_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M2: set_nat] :
( ( member_set_nat @ M2 @ A )
=> ( finite_finite_nat @ M2 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_176_finite__Union,axiom,
! [A: set_set_option_nat] :
( ( finite1464753433994532717on_nat @ A )
=> ( ! [M2: set_option_nat] :
( ( member3860231779568403053on_nat @ M2 @ A )
=> ( finite5523153139673422903on_nat @ M2 ) )
=> ( finite5523153139673422903on_nat @ ( comple3326054718015411497on_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_177_finite__Union,axiom,
! [A: set_se3873067930692246379at_nat] :
( ( finite5325900196762371532at_nat @ A )
=> ( ! [M2: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ M2 @ A )
=> ( finite6187706683773761046at_nat @ M2 ) )
=> ( finite6187706683773761046at_nat @ ( comple2155544827851854728at_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_178_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_179_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_180_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_181_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_182_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_183_dual__order_Orefl,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_184_dual__order_Orefl,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_185_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_186_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_187_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_188_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_189_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_190_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_191_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_192_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_193_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_194_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_195_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_196_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_197_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_198_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_199_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_200_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_201_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_202_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_203_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_204_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_205_nless__le,axiom,
! [A4: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_206_nless__le,axiom,
! [A4: real,B3: real] :
( ( ~ ( ord_less_real @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_207_nless__le,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A4 @ B3 ) )
= ( ~ ( ord_le4245800335709223507ccount @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_208_nless__le,axiom,
! [A4: int,B3: int] :
( ( ~ ( ord_less_int @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_209_nle__le,axiom,
! [A4: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_210_nle__le,axiom,
! [A4: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A4 @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_211_nle__le,axiom,
! [A4: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_212_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_213_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_214_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_215_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_216_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_217_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_218_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_219_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_220_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_221_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_222_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z2: risk_Free_account] : ( Y3 = Z2 ) )
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_223_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_224_ord__eq__le__trans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( A4 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_225_ord__eq__le__trans,axiom,
! [A4: real,B3: real,C: real] :
( ( A4 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_226_ord__eq__le__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( A4 = B3 )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_227_ord__eq__le__trans,axiom,
! [A4: int,B3: int,C: int] :
( ( A4 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_228_ord__le__eq__trans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_229_ord__le__eq__trans,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_230_ord__le__eq__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_231_ord__le__eq__trans,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_232_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_233_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_234_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_235_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_236_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_237_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_238_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_239_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_240_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_241_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_242_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_243_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_244_order_Oasym,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A4 ) ) ).
% order.asym
thf(fact_245_order_Oasym,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ~ ( ord_le2131251472502387783ccount @ B3 @ A4 ) ) ).
% order.asym
thf(fact_246_order_Oasym,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ~ ( ord_less_real @ B3 @ A4 ) ) ).
% order.asym
thf(fact_247_order_Oasym,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ~ ( ord_less_int @ B3 @ A4 ) ) ).
% order.asym
thf(fact_248_order_Otrans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% order.trans
thf(fact_249_order_Otrans,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% order.trans
thf(fact_250_order_Otrans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% order.trans
thf(fact_251_order_Otrans,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% order.trans
thf(fact_252_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_253_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_254_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_255_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_256_ord__eq__less__trans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( A4 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_257_ord__eq__less__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( A4 = B3 )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_258_ord__eq__less__trans,axiom,
! [A4: real,B3: real,C: real] :
( ( A4 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_259_ord__eq__less__trans,axiom,
! [A4: int,B3: int,C: int] :
( ( A4 = B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_260_ord__less__eq__trans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_261_ord__less__eq__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_262_ord__less__eq__trans,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_263_ord__less__eq__trans,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_264_less__induct,axiom,
! [P: nat > $o,A4: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A4 ) ) ).
% less_induct
thf(fact_265_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_266_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_267_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_268_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B3: nat] :
( ! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: nat,B4: nat] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_269_linorder__wlog,axiom,
! [P: real > real > $o,A4: real,B3: real] :
( ! [A2: real,B4: real] :
( ( ord_less_eq_real @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: real,B4: real] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_270_linorder__wlog,axiom,
! [P: int > int > $o,A4: int,B3: int] :
( ! [A2: int,B4: int] :
( ( ord_less_eq_int @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: int,B4: int] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_271_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_272_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_273_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_274_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_275_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_276_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_277_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_278_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z2: risk_Free_account] : ( Y3 = Z2 ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_279_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_280_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_281_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_282_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ~ ( ord_le4245800335709223507ccount @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_283_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_284_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_285_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_286_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_287_dual__order_Oantisym,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_288_dual__order_Oantisym,axiom,
! [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_289_dual__order_Oantisym,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B3 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_290_dual__order_Oantisym,axiom,
! [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_291_dual__order_Oasym,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_292_dual__order_Oasym,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B3 @ A4 )
=> ~ ( ord_le2131251472502387783ccount @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_293_dual__order_Oasym,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ~ ( ord_less_real @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_294_dual__order_Oasym,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ~ ( ord_less_int @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_295_dual__order_Otrans,axiom,
! [B3: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_296_dual__order_Otrans,axiom,
! [B3: real,A4: real,C: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_297_dual__order_Otrans,axiom,
! [B3: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B3 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C @ B3 )
=> ( ord_le4245800335709223507ccount @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_298_dual__order_Otrans,axiom,
! [B3: int,A4: int,C: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_299_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_300_dual__order_Oirrefl,axiom,
! [A4: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_301_dual__order_Oirrefl,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_302_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_303_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_304_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B3: nat] :
( ! [A2: nat,B4: nat] :
( ( ord_less_nat @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B4: nat] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_305_linorder__less__wlog,axiom,
! [P: real > real > $o,A4: real,B3: real] :
( ! [A2: real,B4: real] :
( ( ord_less_real @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B4: real] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_306_linorder__less__wlog,axiom,
! [P: int > int > $o,A4: int,B3: int] :
( ! [A2: int,B4: int] :
( ( ord_less_int @ A2 @ B4 )
=> ( P @ A2 @ B4 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B4: int] :
( ( P @ B4 @ A2 )
=> ( P @ A2 @ B4 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_307_order_Ostrict__trans,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_308_order_Ostrict__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_309_order_Ostrict__trans,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_310_order_Ostrict__trans,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_311_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_312_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_313_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_314_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_315_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_316_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_317_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_318_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_319_order_Ostrict__trans1,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_320_order_Ostrict__trans1,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_321_order_Ostrict__trans1,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_322_order_Ostrict__trans1,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_323_order_Ostrict__trans2,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_324_order_Ostrict__trans2,axiom,
! [A4: real,B3: real,C: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_325_order_Ostrict__trans2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_326_order_Ostrict__trans2,axiom,
! [A4: int,B3: int,C: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_327_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_328_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_329_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ~ ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_330_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_331_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_332_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_333_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_334_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_335_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_336_dual__order_Ostrict__trans,axiom,
! [B3: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_337_dual__order_Ostrict__trans,axiom,
! [B3: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B3 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C @ B3 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_338_dual__order_Ostrict__trans,axiom,
! [B3: real,A4: real,C: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_339_dual__order_Ostrict__trans,axiom,
! [B3: int,A4: int,C: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_340_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_341_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_342_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_343_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_344_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_345_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_346_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_347_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_348_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_349_order_Ostrict__implies__not__eq,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_350_order_Ostrict__implies__not__eq,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_351_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_352_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_353_dual__order_Ostrict__trans1,axiom,
! [B3: real,A4: real,C: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_354_dual__order_Ostrict__trans1,axiom,
! [B3: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B3 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C @ B3 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B3: int,A4: int,C: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_357_dual__order_Ostrict__trans2,axiom,
! [B3: real,A4: real,C: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_358_dual__order_Ostrict__trans2,axiom,
! [B3: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B3 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C @ B3 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_359_dual__order_Ostrict__trans2,axiom,
! [B3: int,A4: int,C: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_360_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_361_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_362_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ~ ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_363_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_364_order_Ostrict__implies__order,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ord_less_eq_nat @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_365_order_Ostrict__implies__order,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ord_less_eq_real @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_366_order_Ostrict__implies__order,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ord_le4245800335709223507ccount @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_367_order_Ostrict__implies__order,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ord_less_eq_int @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_368_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_369_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_370_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_371_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_372_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ord_less_eq_nat @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_373_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ord_less_eq_real @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_374_dual__order_Ostrict__implies__order,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B3 @ A4 )
=> ( ord_le4245800335709223507ccount @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_375_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ord_less_eq_int @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_376_antisym,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_377_antisym,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_378_antisym,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_379_antisym,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_380_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_381_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_382_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z2: risk_Free_account] : ( Y3 = Z2 ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_383_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_384_order__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_385_order__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_386_order__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_387_order__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_388_order__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst1,axiom,
! [A4: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_390_order__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_391_order__subst1,axiom,
! [A4: real,F: int > real,B3: int,C: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_392_order__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B3: real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_395_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_396_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_397_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_398_order__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_399_order__subst2,axiom,
! [A4: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_400_order__subst2,axiom,
! [A4: real,B3: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_401_order__subst2,axiom,
! [A4: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_402_order__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_403_order__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_404_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_405_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_406_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_407_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_408_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_409_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_410_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_411_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_412_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_413_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_414_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_415_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_416_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_417_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_418_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_419_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_420_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_421_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_422_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_423_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_424_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_425_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_426_ord__eq__le__subst,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_427_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
! [A4: int,F: nat > int,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_429_ord__eq__le__subst,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
! [A4: real,F: real > real,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A4: int,F: real > int,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_436_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_437_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_438_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_439_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_440_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_441_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_442_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_443_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_444_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_446_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_447_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_448_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_449_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_450_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_451_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_452_order__less__asym_H,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_453_order__less__asym_H,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ~ ( ord_le2131251472502387783ccount @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_454_order__less__asym_H,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ~ ( ord_less_real @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_455_order__less__asym_H,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ~ ( ord_less_int @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_456_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_457_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_458_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_459_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_460_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_461_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_462_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_463_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_464_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_465_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_466_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_467_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_468_ord__eq__less__subst,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_469_ord__eq__less__subst,axiom,
! [A4: int,F: nat > int,B3: nat,C: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_470_ord__eq__less__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_471_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_472_ord__eq__less__subst,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_473_ord__eq__less__subst,axiom,
! [A4: int,F: risk_Free_account > int,B3: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_474_ord__eq__less__subst,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_475_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B3: real,C: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_476_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_477_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_478_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_479_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_480_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_481_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_482_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_483_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_484_ord__less__eq__subst,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_485_ord__less__eq__subst,axiom,
! [A4: real,B3: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_486_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_487_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_488_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_489_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_490_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_491_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_492_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_493_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_494_order__less__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_495_order__less__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_496_order__less__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_497_order__less__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_498_order__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_499_order__less__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_500_order__less__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B3: real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_501_order__less__subst1,axiom,
! [A4: risk_Free_account,F: int > risk_Free_account,B3: int,C: int] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_502_order__less__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_503_order__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_504_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_505_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_506_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_507_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_508_order__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_509_order__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_510_order__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_511_order__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_512_order__less__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_513_order__less__subst2,axiom,
! [A4: real,B3: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_514_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_515_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_516_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_517_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_518_order__le__neq__trans,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_nat @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_519_order__le__neq__trans,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_real @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_520_order__le__neq__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_le2131251472502387783ccount @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_521_order__le__neq__trans,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_int @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_522_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_523_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_524_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_525_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_526_order__neq__le__trans,axiom,
! [A4: nat,B3: nat] :
( ( A4 != B3 )
=> ( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ord_less_nat @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_527_order__neq__le__trans,axiom,
! [A4: real,B3: real] :
( ( A4 != B3 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ord_less_real @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_528_order__neq__le__trans,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( A4 != B3 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ord_le2131251472502387783ccount @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_529_order__neq__le__trans,axiom,
! [A4: int,B3: int] :
( ( A4 != B3 )
=> ( ( ord_less_eq_int @ A4 @ B3 )
=> ( ord_less_int @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_order__le__less__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_546_order__le__less__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_547_order__le__less__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_548_order__le__less__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_549_order__le__less__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_550_order__le__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_551_order__le__less__subst1,axiom,
! [A4: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_552_order__le__less__subst1,axiom,
! [A4: real,F: int > real,B3: int,C: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_553_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_554_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_le2131251472502387783ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_555_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_556_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_557_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_558_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_559_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_560_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_561_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_562_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_563_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_564_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_565_order__less__le__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_566_order__less__le__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_567_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B3: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_568_order__less__le__subst1,axiom,
! [A4: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_569_order__less__le__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_570_order__less__le__subst1,axiom,
! [A4: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_571_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B3: real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_572_order__less__le__subst1,axiom,
! [A4: int,F: real > int,B3: real,C: real] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_573_order__less__le__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_574_order__less__le__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B3: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_le4245800335709223507ccount @ B3 @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_575_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_576_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_577_order__less__le__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_578_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_579_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_580_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_581_order__less__le__subst2,axiom,
! [A4: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_582_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > real,C: real] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_583_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_584_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B3: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B3 ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_585_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_586_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_587_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_588_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_589_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_590_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_591_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_592_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_593_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_594_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_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_605_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_606_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_607_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_608_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_609_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_610_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_611_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_612_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_613_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_614_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_615_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_616_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_617_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_618_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_619_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_620_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( M3 != N ) ) ) ) ).
% nat_less_le
thf(fact_621_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_622_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_623_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_624_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_625_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_626_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_627_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_628_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_629_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_630_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_631_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_632_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_633_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_634_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_635_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_636_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_637_bij__betw__finite,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_638_bij__betw__finite,axiom,
! [F: nat > option_nat,A: set_nat,B: set_option_nat] :
( ( bij_be1519204712286531816on_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite5523153139673422903on_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_639_bij__betw__finite,axiom,
! [F: nat > sum_sum_nat_nat,A: set_nat,B: set_Sum_sum_nat_nat] :
( ( bij_be4790990086886966983at_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite6187706683773761046at_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_640_bij__betw__finite,axiom,
! [F: option_nat > nat,A: set_option_nat,B: set_nat] :
( ( bij_be5439767378562641512at_nat @ F @ A @ B )
=> ( ( finite5523153139673422903on_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_641_bij__betw__finite,axiom,
! [F: option_nat > option_nat,A: set_option_nat,B: set_option_nat] :
( ( bij_be5006537294939589560on_nat @ F @ A @ B )
=> ( ( finite5523153139673422903on_nat @ A )
= ( finite5523153139673422903on_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_642_bij__betw__finite,axiom,
! [F: option_nat > sum_sum_nat_nat,A: set_option_nat,B: set_Sum_sum_nat_nat] :
( ( bij_be85274426780410263at_nat @ F @ A @ B )
=> ( ( finite5523153139673422903on_nat @ A )
= ( finite6187706683773761046at_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_643_bij__betw__finite,axiom,
! [F: sum_sum_nat_nat > nat,A: set_Sum_sum_nat_nat,B: set_nat] :
( ( bij_be5432664580149595207at_nat @ F @ A @ B )
=> ( ( finite6187706683773761046at_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_644_bij__betw__finite,axiom,
! [F: sum_sum_nat_nat > option_nat,A: set_Sum_sum_nat_nat,B: set_option_nat] :
( ( bij_be9078631923599912727on_nat @ F @ A @ B )
=> ( ( finite6187706683773761046at_nat @ A )
= ( finite5523153139673422903on_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_645_bij__betw__finite,axiom,
! [F: sum_sum_nat_nat > sum_sum_nat_nat,A: set_Sum_sum_nat_nat,B: set_Sum_sum_nat_nat] :
( ( bij_be6758699879772063990at_nat @ F @ A @ B )
=> ( ( finite6187706683773761046at_nat @ A )
= ( finite6187706683773761046at_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_646_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_647_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_648_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_649_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_650_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_651_top_Oextremum__strict,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A4 ) ).
% top.extremum_strict
thf(fact_652_top_Oextremum__strict,axiom,
! [A4: set_option_nat] :
~ ( ord_le1792839605950587050on_nat @ top_to8920198386146353926on_nat @ A4 ) ).
% top.extremum_strict
thf(fact_653_top_Oextremum__strict,axiom,
! [A4: set_Sum_sum_nat_nat] :
~ ( ord_le2904074325318523657at_nat @ top_to6661820994512907621at_nat @ A4 ) ).
% top.extremum_strict
thf(fact_654_top_Oextremum__strict,axiom,
! [A4: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A4 ) ).
% top.extremum_strict
thf(fact_655_top_Oextremum__strict,axiom,
! [A4: set_nat_real] :
~ ( ord_le3527643927072297637t_real @ top_top_set_nat_real @ A4 ) ).
% top.extremum_strict
thf(fact_656_top_Onot__eq__extremum,axiom,
! [A4: set_nat] :
( ( A4 != top_top_set_nat )
= ( ord_less_set_nat @ A4 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_657_top_Onot__eq__extremum,axiom,
! [A4: set_option_nat] :
( ( A4 != top_to8920198386146353926on_nat )
= ( ord_le1792839605950587050on_nat @ A4 @ top_to8920198386146353926on_nat ) ) ).
% top.not_eq_extremum
thf(fact_658_top_Onot__eq__extremum,axiom,
! [A4: set_Sum_sum_nat_nat] :
( ( A4 != top_to6661820994512907621at_nat )
= ( ord_le2904074325318523657at_nat @ A4 @ top_to6661820994512907621at_nat ) ) ).
% top.not_eq_extremum
thf(fact_659_top_Onot__eq__extremum,axiom,
! [A4: set_real] :
( ( A4 != top_top_set_real )
= ( ord_less_set_real @ A4 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_660_top_Onot__eq__extremum,axiom,
! [A4: set_nat_real] :
( ( A4 != top_top_set_nat_real )
= ( ord_le3527643927072297637t_real @ A4 @ top_top_set_nat_real ) ) ).
% top.not_eq_extremum
thf(fact_661_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_662_top__greatest,axiom,
! [A4: set_option_nat] : ( ord_le6937355464348597430on_nat @ A4 @ top_to8920198386146353926on_nat ) ).
% top_greatest
thf(fact_663_top__greatest,axiom,
! [A4: set_Sum_sum_nat_nat] : ( ord_le5967974642961909525at_nat @ A4 @ top_to6661820994512907621at_nat ) ).
% top_greatest
thf(fact_664_top__greatest,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ top_top_set_real ) ).
% top_greatest
thf(fact_665_top__greatest,axiom,
! [A4: set_nat_real] : ( ord_le2908806416726583473t_real @ A4 @ top_top_set_nat_real ) ).
% top_greatest
thf(fact_666_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_667_top_Oextremum__unique,axiom,
! [A4: set_option_nat] :
( ( ord_le6937355464348597430on_nat @ top_to8920198386146353926on_nat @ A4 )
= ( A4 = top_to8920198386146353926on_nat ) ) ).
% top.extremum_unique
thf(fact_668_top_Oextremum__unique,axiom,
! [A4: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ top_to6661820994512907621at_nat @ A4 )
= ( A4 = top_to6661820994512907621at_nat ) ) ).
% top.extremum_unique
thf(fact_669_top_Oextremum__unique,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
= ( A4 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_670_top_Oextremum__unique,axiom,
! [A4: set_nat_real] :
( ( ord_le2908806416726583473t_real @ top_top_set_nat_real @ A4 )
= ( A4 = top_top_set_nat_real ) ) ).
% top.extremum_unique
thf(fact_671_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_672_top_Oextremum__uniqueI,axiom,
! [A4: set_option_nat] :
( ( ord_le6937355464348597430on_nat @ top_to8920198386146353926on_nat @ A4 )
=> ( A4 = top_to8920198386146353926on_nat ) ) ).
% top.extremum_uniqueI
thf(fact_673_top_Oextremum__uniqueI,axiom,
! [A4: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ top_to6661820994512907621at_nat @ A4 )
=> ( A4 = top_to6661820994512907621at_nat ) ) ).
% top.extremum_uniqueI
thf(fact_674_top_Oextremum__uniqueI,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
=> ( A4 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_675_top_Oextremum__uniqueI,axiom,
! [A4: set_nat_real] :
( ( ord_le2908806416726583473t_real @ top_top_set_nat_real @ A4 )
=> ( A4 = top_top_set_nat_real ) ) ).
% top.extremum_uniqueI
thf(fact_676_finite__has__maximal2,axiom,
! [A: set_nat_real,A4: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A4 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ A4 @ X2 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_677_finite__has__maximal2,axiom,
! [A: set_nat,A4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A4 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A4 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_678_finite__has__maximal2,axiom,
! [A: set_real,A4: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A4 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ A4 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_679_finite__has__maximal2,axiom,
! [A: set_Ri1641125681238393385ccount,A4: risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ( member5612106785598075018ccount @ A4 @ A )
=> ? [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
& ( ord_le4245800335709223507ccount @ A4 @ X2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A )
=> ( ( ord_le4245800335709223507ccount @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_680_finite__has__maximal2,axiom,
! [A: set_int,A4: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A4 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A4 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_681_finite__has__minimal2,axiom,
! [A: set_nat_real,A4: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A4 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ X2 @ A4 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_682_finite__has__minimal2,axiom,
! [A: set_nat,A4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A4 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_683_finite__has__minimal2,axiom,
! [A: set_real,A4: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A4 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ X2 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_684_finite__has__minimal2,axiom,
! [A: set_Ri1641125681238393385ccount,A4: risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ( member5612106785598075018ccount @ A4 @ A )
=> ? [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
& ( ord_le4245800335709223507ccount @ X2 @ A4 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A )
=> ( ( ord_le4245800335709223507ccount @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_685_finite__has__minimal2,axiom,
! [A: set_int,A4: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A4 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_686_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_687_finite__UnionD,axiom,
! [A: set_set_option_nat] :
( ( finite5523153139673422903on_nat @ ( comple3326054718015411497on_nat @ A ) )
=> ( finite1464753433994532717on_nat @ A ) ) ).
% finite_UnionD
thf(fact_688_finite__UnionD,axiom,
! [A: set_se3873067930692246379at_nat] :
( ( finite6187706683773761046at_nat @ ( comple2155544827851854728at_nat @ A ) )
=> ( finite5325900196762371532at_nat @ A ) ) ).
% finite_UnionD
thf(fact_689_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_690_Sup__UNIV,axiom,
( ( comple3326054718015411497on_nat @ top_to3692820865130894140on_nat )
= top_to8920198386146353926on_nat ) ).
% Sup_UNIV
thf(fact_691_Sup__UNIV,axiom,
( ( comple2155544827851854728at_nat @ top_to6951743751474147867at_nat )
= top_to6661820994512907621at_nat ) ).
% Sup_UNIV
thf(fact_692_Sup__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Sup_UNIV
thf(fact_693_Sup__UNIV,axiom,
( ( comple6299889548703266724t_real @ top_to1863808837862421559t_real )
= top_top_set_nat_real ) ).
% Sup_UNIV
thf(fact_694_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_695_Union__UNIV,axiom,
( ( comple3326054718015411497on_nat @ top_to3692820865130894140on_nat )
= top_to8920198386146353926on_nat ) ).
% Union_UNIV
thf(fact_696_Union__UNIV,axiom,
( ( comple2155544827851854728at_nat @ top_to6951743751474147867at_nat )
= top_to6661820994512907621at_nat ) ).
% Union_UNIV
thf(fact_697_Union__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Union_UNIV
thf(fact_698_Union__UNIV,axiom,
( ( comple6299889548703266724t_real @ top_to1863808837862421559t_real )
= top_top_set_nat_real ) ).
% Union_UNIV
thf(fact_699_finite__imp__Sup__less,axiom,
! [X6: set_nat,X: nat,A4: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_nat @ X2 @ A4 ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A4 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_700_finite__imp__Sup__less,axiom,
! [X6: set_real,X: real,A4: real] :
( ( finite_finite_real @ X6 )
=> ( ( member_real @ X @ X6 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ( ord_less_real @ X2 @ A4 ) )
=> ( ord_less_real @ ( comple1385675409528146559p_real @ X6 ) @ A4 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_701_finite__imp__Sup__less,axiom,
! [X6: set_int,X: int,A4: int] :
( ( finite_finite_int @ X6 )
=> ( ( member_int @ X @ X6 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ( ord_less_int @ X2 @ A4 ) )
=> ( ord_less_int @ ( complete_Sup_Sup_int @ X6 ) @ A4 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_702_le__cSup__finite,axiom,
! [X6: set_nat,X: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_703_le__cSup__finite,axiom,
! [X6: set_real,X: real] :
( ( finite_finite_real @ X6 )
=> ( ( member_real @ X @ X6 )
=> ( ord_less_eq_real @ X @ ( comple1385675409528146559p_real @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_704_le__cSup__finite,axiom,
! [X6: set_int,X: int] :
( ( finite_finite_int @ X6 )
=> ( ( member_int @ X @ X6 )
=> ( ord_less_eq_int @ X @ ( complete_Sup_Sup_int @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_705_Union__iff,axiom,
! [A: nat > real,C2: set_set_nat_real] :
( ( member_nat_real @ A @ ( comple6299889548703266724t_real @ C2 ) )
= ( ? [X3: set_nat_real] :
( ( member_set_nat_real @ X3 @ C2 )
& ( member_nat_real @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_706_Union__iff,axiom,
! [A: real,C2: set_set_real] :
( ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) )
= ( ? [X3: set_real] :
( ( member_set_real @ X3 @ C2 )
& ( member_real @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_707_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C2 )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_708_UnionI,axiom,
! [X6: set_nat_real,C2: set_set_nat_real,A: nat > real] :
( ( member_set_nat_real @ X6 @ C2 )
=> ( ( member_nat_real @ A @ X6 )
=> ( member_nat_real @ A @ ( comple6299889548703266724t_real @ C2 ) ) ) ) ).
% UnionI
thf(fact_709_UnionI,axiom,
! [X6: set_real,C2: set_set_real,A: real] :
( ( member_set_real @ X6 @ C2 )
=> ( ( member_real @ A @ X6 )
=> ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) ) ) ) ).
% UnionI
thf(fact_710_UnionI,axiom,
! [X6: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X6 @ C2 )
=> ( ( member_nat @ A @ X6 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_711_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A3 ) @ ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_order_embed
thf(fact_712_net__asset__value__mono,axiom,
! [Alpha2: risk_Free_account,Beta: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha2 @ Beta )
=> ( ord_less_eq_real @ ( risk_F2906766666041932210_value @ Alpha2 ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_mono
thf(fact_713_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_714_finite__psubset__induct,axiom,
! [A: set_option_nat,P: set_option_nat > $o] :
( ( finite5523153139673422903on_nat @ A )
=> ( ! [A6: set_option_nat] :
( ( finite5523153139673422903on_nat @ A6 )
=> ( ! [B5: set_option_nat] :
( ( ord_le1792839605950587050on_nat @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_715_finite__psubset__induct,axiom,
! [A: set_Sum_sum_nat_nat,P: set_Sum_sum_nat_nat > $o] :
( ( finite6187706683773761046at_nat @ A )
=> ( ! [A6: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ A6 )
=> ( ! [B5: set_Sum_sum_nat_nat] :
( ( ord_le2904074325318523657at_nat @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_716_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_717_subset__UNIV,axiom,
! [A: set_option_nat] : ( ord_le6937355464348597430on_nat @ A @ top_to8920198386146353926on_nat ) ).
% subset_UNIV
thf(fact_718_subset__UNIV,axiom,
! [A: set_Sum_sum_nat_nat] : ( ord_le5967974642961909525at_nat @ A @ top_to6661820994512907621at_nat ) ).
% subset_UNIV
thf(fact_719_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_720_subset__UNIV,axiom,
! [A: set_nat_real] : ( ord_le2908806416726583473t_real @ A @ top_top_set_nat_real ) ).
% subset_UNIV
thf(fact_721_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_722_rev__finite__subset,axiom,
! [B: set_option_nat,A: set_option_nat] :
( ( finite5523153139673422903on_nat @ B )
=> ( ( ord_le6937355464348597430on_nat @ A @ B )
=> ( finite5523153139673422903on_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_723_rev__finite__subset,axiom,
! [B: set_Sum_sum_nat_nat,A: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ B )
=> ( ( ord_le5967974642961909525at_nat @ A @ B )
=> ( finite6187706683773761046at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_724_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_725_infinite__super,axiom,
! [S2: set_option_nat,T2: set_option_nat] :
( ( ord_le6937355464348597430on_nat @ S2 @ T2 )
=> ( ~ ( finite5523153139673422903on_nat @ S2 )
=> ~ ( finite5523153139673422903on_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_726_infinite__super,axiom,
! [S2: set_Sum_sum_nat_nat,T2: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ S2 @ T2 )
=> ( ~ ( finite6187706683773761046at_nat @ S2 )
=> ~ ( finite6187706683773761046at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_727_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_728_finite__subset,axiom,
! [A: set_option_nat,B: set_option_nat] :
( ( ord_le6937355464348597430on_nat @ A @ B )
=> ( ( finite5523153139673422903on_nat @ B )
=> ( finite5523153139673422903on_nat @ A ) ) ) ).
% finite_subset
thf(fact_729_finite__subset,axiom,
! [A: set_Sum_sum_nat_nat,B: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ A @ B )
=> ( ( finite6187706683773761046at_nat @ B )
=> ( finite6187706683773761046at_nat @ A ) ) ) ).
% finite_subset
thf(fact_730_lists__infinite__bij__betw__types,axiom,
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ? [F3: list_nat > nat] : ( bij_be8532844293280997160at_nat @ F3 @ top_top_set_list_nat @ top_top_set_nat ) ) ).
% lists_infinite_bij_betw_types
thf(fact_731_lists__infinite__bij__betw__types,axiom,
( ~ ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ? [F3: list_option_nat > option_nat] : ( bij_be8029046853998793672on_nat @ F3 @ top_to4146629662433032470on_nat @ top_to8920198386146353926on_nat ) ) ).
% lists_infinite_bij_betw_types
thf(fact_732_lists__infinite__bij__betw__types,axiom,
( ~ ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
=> ? [F3: list_Sum_sum_nat_nat > sum_sum_nat_nat] : ( bij_be1420823545580895622at_nat @ F3 @ top_to7612388123161931765at_nat @ top_to6661820994512907621at_nat ) ) ).
% lists_infinite_bij_betw_types
thf(fact_733_lists__infinite__bij__betw__types,axiom,
( ~ ( finite_finite_real @ top_top_set_real )
=> ? [F3: list_real > real] : ( bij_be9064865367860874976l_real @ F3 @ top_to7938183770042617506t_real @ top_top_set_real ) ) ).
% lists_infinite_bij_betw_types
thf(fact_734_lists__infinite__bij__betw__types,axiom,
( ~ ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ? [F3: list_nat_real > nat > real] : ( bij_be48681600348046398t_real @ F3 @ top_to8431753751349480721t_real @ top_top_set_nat_real ) ) ).
% lists_infinite_bij_betw_types
thf(fact_735_ex__gt__or__lt,axiom,
! [A4: real] :
? [B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( ord_less_real @ B4 @ A4 ) ) ).
% ex_gt_or_lt
thf(fact_736_UnionE,axiom,
! [A: nat > real,C2: set_set_nat_real] :
( ( member_nat_real @ A @ ( comple6299889548703266724t_real @ C2 ) )
=> ~ ! [X7: set_nat_real] :
( ( member_nat_real @ A @ X7 )
=> ~ ( member_set_nat_real @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_737_UnionE,axiom,
! [A: real,C2: set_set_real] :
( ( member_real @ A @ ( comple3096694443085538997t_real @ C2 ) )
=> ~ ! [X7: set_real] :
( ( member_real @ A @ X7 )
=> ~ ( member_set_real @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_738_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_739_complete__interval,axiom,
! [A4: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A4 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X8: nat] :
( ( ( ord_less_eq_nat @ A4 @ X8 )
& ( ord_less_nat @ X8 @ C3 ) )
=> ( P @ X8 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A4 @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_740_complete__interval,axiom,
! [A4: real,B3: real,P: real > $o] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A4 @ C3 )
& ( ord_less_eq_real @ C3 @ B3 )
& ! [X8: real] :
( ( ( ord_less_eq_real @ A4 @ X8 )
& ( ord_less_real @ X8 @ C3 ) )
=> ( P @ X8 ) )
& ! [D: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_real @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_741_complete__interval,axiom,
! [A4: int,B3: int,P: int > $o] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A4 @ C3 )
& ( ord_less_eq_int @ C3 @ B3 )
& ! [X8: int] :
( ( ( ord_less_eq_int @ A4 @ X8 )
& ( ord_less_int @ X8 @ C3 ) )
=> ( P @ X8 ) )
& ! [D: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A4 @ X2 )
& ( ord_less_int @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_742_cSup__eq__maximum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_743_cSup__eq__maximum,axiom,
! [Z: real,X6: set_real] :
( ( member_real @ Z @ X6 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_744_cSup__eq__maximum,axiom,
! [Z: int,X6: set_int] :
( ( member_int @ Z @ X6 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_745_cSup__eq,axiom,
! [X6: set_real,A4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ( ord_less_eq_real @ X2 @ A4 ) )
=> ( ! [Y2: real] :
( ! [X8: real] :
( ( member_real @ X8 @ X6 )
=> ( ord_less_eq_real @ X8 @ Y2 ) )
=> ( ord_less_eq_real @ A4 @ Y2 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= A4 ) ) ) ).
% cSup_eq
thf(fact_746_cSup__eq,axiom,
! [X6: set_int,A4: int] :
( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ( ord_less_eq_int @ X2 @ A4 ) )
=> ( ! [Y2: int] :
( ! [X8: int] :
( ( member_int @ X8 @ X6 )
=> ( ord_less_eq_int @ X8 @ Y2 ) )
=> ( ord_less_eq_int @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup_int @ X6 )
= A4 ) ) ) ).
% cSup_eq
thf(fact_747_finite__subset__Union,axiom,
! [A: set_nat,B6: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B6 ) )
=> ~ ! [F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( ord_le6893508408891458716et_nat @ F4 @ B6 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_748_finite__subset__Union,axiom,
! [A: set_option_nat,B6: set_set_option_nat] :
( ( finite5523153139673422903on_nat @ A )
=> ( ( ord_le6937355464348597430on_nat @ A @ ( comple3326054718015411497on_nat @ B6 ) )
=> ~ ! [F4: set_set_option_nat] :
( ( finite1464753433994532717on_nat @ F4 )
=> ( ( ord_le2998974513579896044on_nat @ F4 @ B6 )
=> ~ ( ord_le6937355464348597430on_nat @ A @ ( comple3326054718015411497on_nat @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_749_finite__subset__Union,axiom,
! [A: set_Sum_sum_nat_nat,B6: set_se3873067930692246379at_nat] :
( ( finite6187706683773761046at_nat @ A )
=> ( ( ord_le5967974642961909525at_nat @ A @ ( comple2155544827851854728at_nat @ B6 ) )
=> ~ ! [F4: set_se3873067930692246379at_nat] :
( ( finite5325900196762371532at_nat @ F4 )
=> ( ( ord_le3495481059733392331at_nat @ F4 @ B6 )
=> ~ ( ord_le5967974642961909525at_nat @ A @ ( comple2155544827851854728at_nat @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_750_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_751_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_752_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_753_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_754_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X2: nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_755_involuntory__imp__bij,axiom,
! [F: option_nat > option_nat] :
( ! [X2: option_nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be5006537294939589560on_nat @ F @ top_to8920198386146353926on_nat @ top_to8920198386146353926on_nat ) ) ).
% involuntory_imp_bij
thf(fact_756_involuntory__imp__bij,axiom,
! [F: sum_sum_nat_nat > sum_sum_nat_nat] :
( ! [X2: sum_sum_nat_nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be6758699879772063990at_nat @ F @ top_to6661820994512907621at_nat @ top_to6661820994512907621at_nat ) ) ).
% involuntory_imp_bij
thf(fact_757_involuntory__imp__bij,axiom,
! [F: real > real] :
( ! [X2: real] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_real_real @ F @ top_top_set_real @ top_top_set_real ) ) ).
% involuntory_imp_bij
thf(fact_758_involuntory__imp__bij,axiom,
! [F: ( nat > real ) > nat > real] :
( ! [X2: nat > real] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be8325372660175221806t_real @ F @ top_top_set_nat_real @ top_top_set_nat_real ) ) ).
% involuntory_imp_bij
thf(fact_759_bij__pointE,axiom,
! [F: nat > nat,Y: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_760_bij__pointE,axiom,
! [F: nat > real,Y: real] :
( ( bij_betw_nat_real @ F @ top_top_set_nat @ top_top_set_real )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_761_bij__pointE,axiom,
! [F: real > nat,Y: nat] :
( ( bij_betw_real_nat @ F @ top_top_set_real @ top_top_set_nat )
=> ~ ! [X2: real] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: real] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_762_bij__pointE,axiom,
! [F: real > real,Y: real] :
( ( bij_betw_real_real @ F @ top_top_set_real @ top_top_set_real )
=> ~ ! [X2: real] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: real] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_763_bij__pointE,axiom,
! [F: nat > option_nat,Y: option_nat] :
( ( bij_be1519204712286531816on_nat @ F @ top_top_set_nat @ top_to8920198386146353926on_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_764_bij__pointE,axiom,
! [F: option_nat > nat,Y: nat] :
( ( bij_be5439767378562641512at_nat @ F @ top_to8920198386146353926on_nat @ top_top_set_nat )
=> ~ ! [X2: option_nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: option_nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_765_bij__pointE,axiom,
! [F: option_nat > real,Y: real] :
( ( bij_be5928300281789875524t_real @ F @ top_to8920198386146353926on_nat @ top_top_set_real )
=> ~ ! [X2: option_nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: option_nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_766_bij__pointE,axiom,
! [F: real > option_nat,Y: option_nat] :
( ( bij_be2388235277519367748on_nat @ F @ top_top_set_real @ top_to8920198386146353926on_nat )
=> ~ ! [X2: real] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: real] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_767_bij__pointE,axiom,
! [F: nat > sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( bij_be4790990086886966983at_nat @ F @ top_top_set_nat @ top_to6661820994512907621at_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_768_bij__pointE,axiom,
! [F: nat > nat > real,Y: nat > real] :
( ( bij_be4710973223651265507t_real @ F @ top_top_set_nat @ top_top_set_nat_real )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_769_subsetI,axiom,
! [A: set_nat_real,B: set_nat_real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( member_nat_real @ X2 @ B ) )
=> ( ord_le2908806416726583473t_real @ A @ B ) ) ).
% subsetI
thf(fact_770_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_771_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_772_in__mono,axiom,
! [A: set_nat_real,B: set_nat_real,X: nat > real] :
( ( ord_le2908806416726583473t_real @ A @ B )
=> ( ( member_nat_real @ X @ A )
=> ( member_nat_real @ X @ B ) ) ) ).
% in_mono
thf(fact_773_in__mono,axiom,
! [A: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_774_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_775_subsetD,axiom,
! [A: set_nat_real,B: set_nat_real,C: nat > real] :
( ( ord_le2908806416726583473t_real @ A @ B )
=> ( ( member_nat_real @ C @ A )
=> ( member_nat_real @ C @ B ) ) ) ).
% subsetD
thf(fact_776_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_777_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_778_psubsetD,axiom,
! [A: set_nat_real,B: set_nat_real,C: nat > real] :
( ( ord_le3527643927072297637t_real @ A @ B )
=> ( ( member_nat_real @ C @ A )
=> ( member_nat_real @ C @ B ) ) ) ).
% psubsetD
thf(fact_779_psubsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% psubsetD
thf(fact_780_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_781_subset__eq,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B7: set_nat_real] :
! [X3: nat > real] :
( ( member_nat_real @ X3 @ A5 )
=> ( member_nat_real @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_782_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B7: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( member_real @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_783_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_784_subset__iff,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B7: set_nat_real] :
! [T3: nat > real] :
( ( member_nat_real @ T3 @ A5 )
=> ( member_nat_real @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_785_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B7: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A5 )
=> ( member_real @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_786_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_787_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_788_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X8: nat] :
( ( P @ X8 )
=> ( ord_less_eq_nat @ X8 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_789_bij__betw__iff__bijections,axiom,
( bij_be8325372660175221806t_real
= ( ^ [F5: ( nat > real ) > nat > real,A5: set_nat_real,B7: set_nat_real] :
? [G: ( nat > real ) > nat > real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A5 )
=> ( ( member_nat_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > real] :
( ( member_nat_real @ X3 @ B7 )
=> ( ( member_nat_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_790_bij__betw__iff__bijections,axiom,
( bij_be1988628644077263679t_real
= ( ^ [F5: real > nat > real,A5: set_real,B7: set_nat_real] :
? [G: ( nat > real ) > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( ( member_nat_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > real] :
( ( member_nat_real @ X3 @ B7 )
=> ( ( member_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_791_bij__betw__iff__bijections,axiom,
( bij_be4710973223651265507t_real
= ( ^ [F5: nat > nat > real,A5: set_nat,B7: set_nat_real] :
? [G: ( nat > real ) > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_nat_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > real] :
( ( member_nat_real @ X3 @ B7 )
=> ( ( member_nat @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_792_bij__betw__iff__bijections,axiom,
( bij_be457252194604027711l_real
= ( ^ [F5: ( nat > real ) > real,A5: set_nat_real,B7: set_real] :
? [G: real > nat > real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A5 )
=> ( ( member_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: real] :
( ( member_real @ X3 @ B7 )
=> ( ( member_nat_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_793_bij__betw__iff__bijections,axiom,
( bij_betw_real_real
= ( ^ [F5: real > real,A5: set_real,B7: set_real] :
? [G: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( ( member_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: real] :
( ( member_real @ X3 @ B7 )
=> ( ( member_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_794_bij__betw__iff__bijections,axiom,
( bij_betw_nat_real
= ( ^ [F5: nat > real,A5: set_nat,B7: set_real] :
? [G: real > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_real @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: real] :
( ( member_real @ X3 @ B7 )
=> ( ( member_nat @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_795_bij__betw__iff__bijections,axiom,
( bij_be2189776203206559203al_nat
= ( ^ [F5: ( nat > real ) > nat,A5: set_nat_real,B7: set_nat] :
? [G: nat > nat > real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A5 )
=> ( ( member_nat @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B7 )
=> ( ( member_nat_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_796_bij__betw__iff__bijections,axiom,
( bij_betw_real_nat
= ( ^ [F5: real > nat,A5: set_real,B7: set_nat] :
? [G: nat > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( ( member_nat @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B7 )
=> ( ( member_real @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_797_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F5: nat > nat,A5: set_nat,B7: set_nat] :
? [G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ ( F5 @ X3 ) @ B7 )
& ( ( G @ ( F5 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B7 )
=> ( ( member_nat @ ( G @ X3 ) @ A5 )
& ( ( F5 @ ( G @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_798_bij__betw__apply,axiom,
! [F: ( nat > real ) > nat > real,A: set_nat_real,B: set_nat_real,A4: nat > real] :
( ( bij_be8325372660175221806t_real @ F @ A @ B )
=> ( ( member_nat_real @ A4 @ A )
=> ( member_nat_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_799_bij__betw__apply,axiom,
! [F: ( nat > real ) > real,A: set_nat_real,B: set_real,A4: nat > real] :
( ( bij_be457252194604027711l_real @ F @ A @ B )
=> ( ( member_nat_real @ A4 @ A )
=> ( member_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_800_bij__betw__apply,axiom,
! [F: ( nat > real ) > nat,A: set_nat_real,B: set_nat,A4: nat > real] :
( ( bij_be2189776203206559203al_nat @ F @ A @ B )
=> ( ( member_nat_real @ A4 @ A )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_801_bij__betw__apply,axiom,
! [F: real > nat > real,A: set_real,B: set_nat_real,A4: real] :
( ( bij_be1988628644077263679t_real @ F @ A @ B )
=> ( ( member_real @ A4 @ A )
=> ( member_nat_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_802_bij__betw__apply,axiom,
! [F: real > real,A: set_real,B: set_real,A4: real] :
( ( bij_betw_real_real @ F @ A @ B )
=> ( ( member_real @ A4 @ A )
=> ( member_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_803_bij__betw__apply,axiom,
! [F: real > nat,A: set_real,B: set_nat,A4: real] :
( ( bij_betw_real_nat @ F @ A @ B )
=> ( ( member_real @ A4 @ A )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_804_bij__betw__apply,axiom,
! [F: nat > nat > real,A: set_nat,B: set_nat_real,A4: nat] :
( ( bij_be4710973223651265507t_real @ F @ A @ B )
=> ( ( member_nat @ A4 @ A )
=> ( member_nat_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_805_bij__betw__apply,axiom,
! [F: nat > real,A: set_nat,B: set_real,A4: nat] :
( ( bij_betw_nat_real @ F @ A @ B )
=> ( ( member_nat @ A4 @ A )
=> ( member_real @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_806_bij__betw__apply,axiom,
! [F: nat > nat,A: set_nat,B: set_nat,A4: nat] :
( ( bij_betw_nat_nat @ F @ A @ B )
=> ( ( member_nat @ A4 @ A )
=> ( member_nat @ ( F @ A4 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_807_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y4: nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_808_bij__iff,axiom,
! [F: nat > real] :
( ( bij_betw_nat_real @ F @ top_top_set_nat @ top_top_set_real )
= ( ! [X3: real] :
? [Y4: nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_809_bij__iff,axiom,
! [F: real > nat] :
( ( bij_betw_real_nat @ F @ top_top_set_real @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y4: real] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: real] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_810_bij__iff,axiom,
! [F: real > real] :
( ( bij_betw_real_real @ F @ top_top_set_real @ top_top_set_real )
= ( ! [X3: real] :
? [Y4: real] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: real] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_811_bij__iff,axiom,
! [F: nat > option_nat] :
( ( bij_be1519204712286531816on_nat @ F @ top_top_set_nat @ top_to8920198386146353926on_nat )
= ( ! [X3: option_nat] :
? [Y4: nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_812_bij__iff,axiom,
! [F: option_nat > nat] :
( ( bij_be5439767378562641512at_nat @ F @ top_to8920198386146353926on_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y4: option_nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: option_nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_813_bij__iff,axiom,
! [F: option_nat > real] :
( ( bij_be5928300281789875524t_real @ F @ top_to8920198386146353926on_nat @ top_top_set_real )
= ( ! [X3: real] :
? [Y4: option_nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: option_nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_814_bij__iff,axiom,
! [F: real > option_nat] :
( ( bij_be2388235277519367748on_nat @ F @ top_top_set_real @ top_to8920198386146353926on_nat )
= ( ! [X3: option_nat] :
? [Y4: real] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: real] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_815_bij__iff,axiom,
! [F: nat > sum_sum_nat_nat] :
( ( bij_be4790990086886966983at_nat @ F @ top_top_set_nat @ top_to6661820994512907621at_nat )
= ( ! [X3: sum_sum_nat_nat] :
? [Y4: nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_816_bij__iff,axiom,
! [F: nat > nat > real] :
( ( bij_be4710973223651265507t_real @ F @ top_top_set_nat @ top_top_set_nat_real )
= ( ! [X3: nat > real] :
? [Y4: nat] :
( ( ( F @ Y4 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y4 ) ) ) ) ) ).
% bij_iff
thf(fact_817_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_818_just__cash__valid__transfer,axiom,
! [C: real,T: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_819_bijI_H,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: nat] :
? [X8: nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_820_bijI_H,axiom,
! [F: nat > real] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: real] :
? [X8: nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_betw_nat_real @ F @ top_top_set_nat @ top_top_set_real ) ) ) ).
% bijI'
thf(fact_821_bijI_H,axiom,
! [F: real > nat] :
( ! [X2: real,Y2: real] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: nat] :
? [X8: real] :
( Y2
= ( F @ X8 ) )
=> ( bij_betw_real_nat @ F @ top_top_set_real @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_822_bijI_H,axiom,
! [F: real > real] :
( ! [X2: real,Y2: real] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: real] :
? [X8: real] :
( Y2
= ( F @ X8 ) )
=> ( bij_betw_real_real @ F @ top_top_set_real @ top_top_set_real ) ) ) ).
% bijI'
thf(fact_823_bijI_H,axiom,
! [F: nat > option_nat] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: option_nat] :
? [X8: nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_be1519204712286531816on_nat @ F @ top_top_set_nat @ top_to8920198386146353926on_nat ) ) ) ).
% bijI'
thf(fact_824_bijI_H,axiom,
! [F: option_nat > nat] :
( ! [X2: option_nat,Y2: option_nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: nat] :
? [X8: option_nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_be5439767378562641512at_nat @ F @ top_to8920198386146353926on_nat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_825_bijI_H,axiom,
! [F: option_nat > real] :
( ! [X2: option_nat,Y2: option_nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: real] :
? [X8: option_nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_be5928300281789875524t_real @ F @ top_to8920198386146353926on_nat @ top_top_set_real ) ) ) ).
% bijI'
thf(fact_826_bijI_H,axiom,
! [F: real > option_nat] :
( ! [X2: real,Y2: real] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: option_nat] :
? [X8: real] :
( Y2
= ( F @ X8 ) )
=> ( bij_be2388235277519367748on_nat @ F @ top_top_set_real @ top_to8920198386146353926on_nat ) ) ) ).
% bijI'
thf(fact_827_bijI_H,axiom,
! [F: nat > sum_sum_nat_nat] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: sum_sum_nat_nat] :
? [X8: nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_be4790990086886966983at_nat @ F @ top_top_set_nat @ top_to6661820994512907621at_nat ) ) ) ).
% bijI'
thf(fact_828_bijI_H,axiom,
! [F: nat > nat > real] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) )
=> ( ! [Y2: nat > real] :
? [X8: nat] :
( Y2
= ( F @ X8 ) )
=> ( bij_be4710973223651265507t_real @ F @ top_top_set_nat @ top_top_set_nat_real ) ) ) ).
% bijI'
thf(fact_829_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_830_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_831_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_832_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_833_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_834_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_835_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X8 ) ) ).
% minf(8)
thf(fact_836_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X8 ) ) ).
% minf(8)
thf(fact_837_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X8 ) ) ).
% minf(8)
thf(fact_838_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ord_less_eq_nat @ X8 @ T ) ) ).
% minf(6)
thf(fact_839_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ord_less_eq_real @ X8 @ T ) ) ).
% minf(6)
thf(fact_840_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( ord_less_eq_int @ X8 @ T ) ) ).
% minf(6)
thf(fact_841_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ord_less_eq_nat @ T @ X8 ) ) ).
% pinf(8)
thf(fact_842_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ord_less_eq_real @ T @ X8 ) ) ).
% pinf(8)
thf(fact_843_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( ord_less_eq_int @ T @ X8 ) ) ).
% pinf(8)
thf(fact_844_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ~ ( ord_less_eq_nat @ X8 @ T ) ) ).
% pinf(6)
thf(fact_845_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ~ ( ord_less_eq_real @ X8 @ T ) ) ).
% pinf(6)
thf(fact_846_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ~ ( ord_less_eq_int @ X8 @ T ) ) ).
% pinf(6)
thf(fact_847_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_848_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_849_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_850_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_851_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_852_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_853_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(3)
thf(fact_854_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(3)
thf(fact_855_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(3)
thf(fact_856_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(4)
thf(fact_857_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(4)
thf(fact_858_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( X8 != T ) ) ).
% pinf(4)
thf(fact_859_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ~ ( ord_less_nat @ X8 @ T ) ) ).
% pinf(5)
thf(fact_860_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ~ ( ord_less_real @ X8 @ T ) ) ).
% pinf(5)
thf(fact_861_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ~ ( ord_less_int @ X8 @ T ) ) ).
% pinf(5)
thf(fact_862_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ord_less_nat @ T @ X8 ) ) ).
% pinf(7)
thf(fact_863_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ord_less_real @ T @ X8 ) ) ).
% pinf(7)
thf(fact_864_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ Z3 @ X8 )
=> ( ord_less_int @ T @ X8 ) ) ).
% pinf(7)
thf(fact_865_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_866_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_867_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_868_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_869_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_870_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_871_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(3)
thf(fact_872_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(3)
thf(fact_873_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(3)
thf(fact_874_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(4)
thf(fact_875_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(4)
thf(fact_876_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( X8 != T ) ) ).
% minf(4)
thf(fact_877_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ord_less_nat @ X8 @ T ) ) ).
% minf(5)
thf(fact_878_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ord_less_real @ X8 @ T ) ) ).
% minf(5)
thf(fact_879_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ( ord_less_int @ X8 @ T ) ) ).
% minf(5)
thf(fact_880_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ~ ( ord_less_nat @ T @ X8 ) ) ).
% minf(7)
thf(fact_881_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ~ ( ord_less_real @ T @ X8 ) ) ).
% minf(7)
thf(fact_882_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z3 )
=> ~ ( ord_less_int @ T @ X8 ) ) ).
% minf(7)
thf(fact_883_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_884_complete__real,axiom,
! [S2: set_real] :
( ? [X8: real] : ( member_real @ X8 @ S2 )
=> ( ? [Z5: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ? [Y2: real] :
( ! [X8: real] :
( ( member_real @ X8 @ S2 )
=> ( ord_less_eq_real @ X8 @ Y2 ) )
& ! [Z5: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ( ord_less_eq_real @ Y2 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_885_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_886_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_887_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_888_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A7 ) )
= ( ord_less_nat @ A7 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_889_verit__comp__simplify1_I3_J,axiom,
! [B8: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B8 @ A7 ) )
= ( ord_less_real @ A7 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_890_verit__comp__simplify1_I3_J,axiom,
! [B8: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B8 @ A7 ) )
= ( ord_less_int @ A7 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_891_strictly__solvent__non__negative__cash,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha2 ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_892_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_893_top_Oordering__top__axioms,axiom,
orderi2192970301710330341on_nat @ ord_le6937355464348597430on_nat @ ord_le1792839605950587050on_nat @ top_to8920198386146353926on_nat ).
% top.ordering_top_axioms
thf(fact_894_top_Oordering__top__axioms,axiom,
orderi8889600011261982788at_nat @ ord_le5967974642961909525at_nat @ ord_le2904074325318523657at_nat @ top_to6661820994512907621at_nat ).
% top.ordering_top_axioms
thf(fact_895_top_Oordering__top__axioms,axiom,
orderi6545683623190734705t_real @ ord_less_eq_set_real @ ord_less_set_real @ top_top_set_real ).
% top.ordering_top_axioms
thf(fact_896_top_Oordering__top__axioms,axiom,
orderi9168907084944354912t_real @ ord_le2908806416726583473t_real @ ord_le3527643927072297637t_real @ top_top_set_nat_real ).
% top.ordering_top_axioms
thf(fact_897_strictly__solvent__net__asset__value,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F2906766666041932210_value @ Alpha2 ) ) ) ).
% strictly_solvent_net_asset_value
thf(fact_898_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_899_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_900_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_901_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_902_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_903_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_904_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_905_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_906_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_907_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_908_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_909_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_910_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_911_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_912_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_913_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_914_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_915_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_916_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_917_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_918_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_919_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_920_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_921_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_922_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_923_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_924_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_925_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_926_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_927_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_928_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_929_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_930_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_931_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_932_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_933_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_934_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_935_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_936_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_937_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_938_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_939_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_940_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha2: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha2 @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha2 ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_941_verit__comp__simplify1_I2_J,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_942_verit__comp__simplify1_I2_J,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_943_verit__comp__simplify1_I2_J,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_944_verit__comp__simplify1_I2_J,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_945_verit__la__disequality,axiom,
! [A4: nat,B3: nat] :
( ( A4 = B3 )
| ~ ( ord_less_eq_nat @ A4 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_946_verit__la__disequality,axiom,
! [A4: real,B3: real] :
( ( A4 = B3 )
| ~ ( ord_less_eq_real @ A4 @ B3 )
| ~ ( ord_less_eq_real @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_947_verit__la__disequality,axiom,
! [A4: int,B3: int] :
( ( A4 = B3 )
| ~ ( ord_less_eq_int @ A4 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_948_verit__comp__simplify1_I1_J,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_949_verit__comp__simplify1_I1_J,axiom,
! [A4: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_950_verit__comp__simplify1_I1_J,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_951_verit__comp__simplify1_I1_J,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_952_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_953_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_954_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_955_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_956_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_957_verit__la__generic,axiom,
! [A4: int,X: int] :
( ( ord_less_eq_int @ A4 @ X )
| ( A4 = X )
| ( ord_less_eq_int @ X @ A4 ) ) ).
% verit_la_generic
thf(fact_958_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_959_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_960_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_961_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_962_conj__le__cong,axiom,
! [X: int,X10: int,P: $o,P4: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_963_imp__le__cong,axiom,
! [X: int,X10: int,P: $o,P4: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_964_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_965_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_966_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_967_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_968_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_969_neg__equal__iff__equal,axiom,
! [A4: int,B3: int] :
( ( ( uminus_uminus_int @ A4 )
= ( uminus_uminus_int @ B3 ) )
= ( A4 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_970_neg__equal__iff__equal,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A4 )
= ( uminus3377898441596595772ccount @ B3 ) )
= ( A4 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_971_neg__equal__iff__equal,axiom,
! [A4: real,B3: real] :
( ( ( uminus_uminus_real @ A4 )
= ( uminus_uminus_real @ B3 ) )
= ( A4 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_972_add_Oinverse__inverse,axiom,
! [A4: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_973_add_Oinverse__inverse,axiom,
! [A4: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_974_add_Oinverse__inverse,axiom,
! [A4: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_975_neg__le__iff__le,axiom,
! [B3: real,A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_976_neg__le__iff__le,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B3 ) @ ( uminus3377898441596595772ccount @ A4 ) )
= ( ord_le4245800335709223507ccount @ A4 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_977_neg__le__iff__le,axiom,
! [B3: int,A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_978_neg__equal__zero,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= A4 )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_979_neg__equal__zero,axiom,
! [A4: real] :
( ( ( uminus_uminus_real @ A4 )
= A4 )
= ( A4 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_980_equal__neg__zero,axiom,
! [A4: int] :
( ( A4
= ( uminus_uminus_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_981_equal__neg__zero,axiom,
! [A4: real] :
( ( A4
= ( uminus_uminus_real @ A4 ) )
= ( A4 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_982_neg__equal__0__iff__equal,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_983_neg__equal__0__iff__equal,axiom,
! [A4: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A4 )
= zero_z1425366712893667068ccount )
= ( A4 = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_984_neg__equal__0__iff__equal,axiom,
! [A4: real] :
( ( ( uminus_uminus_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_985_neg__0__equal__iff__equal,axiom,
! [A4: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A4 ) )
= ( zero_zero_int = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_986_neg__0__equal__iff__equal,axiom,
! [A4: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A4 ) )
= ( zero_z1425366712893667068ccount = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_987_neg__0__equal__iff__equal,axiom,
! [A4: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A4 ) )
= ( zero_zero_real = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_988_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_989_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_990_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_991_neg__less__iff__less,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_992_neg__less__iff__less,axiom,
! [B3: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B3 ) @ ( uminus3377898441596595772ccount @ A4 ) )
= ( ord_le2131251472502387783ccount @ A4 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_993_neg__less__iff__less,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_994_neg__0__le__iff__le,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_995_neg__0__le__iff__le,axiom,
! [A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A4 ) )
= ( ord_le4245800335709223507ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% neg_0_le_iff_le
thf(fact_996_neg__0__le__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_997_neg__le__0__iff__le,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_998_neg__le__0__iff__le,axiom,
! [A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A4 ) @ zero_z1425366712893667068ccount )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_999_neg__le__0__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_1000_less__eq__neg__nonpos,axiom,
! [A4: real] :
( ( ord_less_eq_real @ A4 @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1001_less__eq__neg__nonpos,axiom,
! [A4: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1002_neg__less__eq__nonneg,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ A4 )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_1003_neg__less__eq__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_1004_less__neg__neg,axiom,
! [A4: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1005_less__neg__neg,axiom,
! [A4: real] :
( ( ord_less_real @ A4 @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1006_neg__less__pos,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_pos
thf(fact_1007_neg__less__pos,axiom,
! [A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ A4 )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% neg_less_pos
thf(fact_1008_neg__0__less__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1009_neg__0__less__iff__less,axiom,
! [A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A4 ) )
= ( ord_le2131251472502387783ccount @ A4 @ zero_z1425366712893667068ccount ) ) ).
% neg_0_less_iff_less
thf(fact_1010_neg__0__less__iff__less,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1011_neg__less__0__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_1012_neg__less__0__iff__less,axiom,
! [A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A4 ) @ zero_z1425366712893667068ccount )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_1013_neg__less__0__iff__less,axiom,
! [A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_1014_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1015_negative__zle,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1016_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1017_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1018_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1019_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1020_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_1021_le__minus__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A4 ) ) ) ).
% le_minus_iff
thf(fact_1022_le__minus__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( uminus3377898441596595772ccount @ B3 ) )
= ( ord_le4245800335709223507ccount @ B3 @ ( uminus3377898441596595772ccount @ A4 ) ) ) ).
% le_minus_iff
thf(fact_1023_le__minus__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_minus_iff
thf(fact_1024_minus__le__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ B3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_1025_minus__le__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A4 ) @ B3 )
= ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B3 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_1026_minus__le__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ B3 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_1027_le__imp__neg__le,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_1028_le__imp__neg__le,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B3 )
=> ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B3 ) @ ( uminus3377898441596595772ccount @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_1029_le__imp__neg__le,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_1030_less__minus__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_int @ B3 @ ( uminus_uminus_int @ A4 ) ) ) ).
% less_minus_iff
thf(fact_1031_less__minus__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( uminus3377898441596595772ccount @ B3 ) )
= ( ord_le2131251472502387783ccount @ B3 @ ( uminus3377898441596595772ccount @ A4 ) ) ) ).
% less_minus_iff
thf(fact_1032_less__minus__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_real @ B3 @ ( uminus_uminus_real @ A4 ) ) ) ).
% less_minus_iff
thf(fact_1033_minus__less__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ B3 )
= ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_1034_minus__less__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A4 ) @ B3 )
= ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B3 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_1035_minus__less__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ B3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_1036_verit__negate__coefficient_I2_J,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1037_verit__negate__coefficient_I2_J,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B3 )
=> ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B3 ) @ ( uminus3377898441596595772ccount @ A4 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1038_verit__negate__coefficient_I2_J,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1039_minus__equation__iff,axiom,
! [A4: int,B3: int] :
( ( ( uminus_uminus_int @ A4 )
= B3 )
= ( ( uminus_uminus_int @ B3 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_1040_minus__equation__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A4 )
= B3 )
= ( ( uminus3377898441596595772ccount @ B3 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_1041_minus__equation__iff,axiom,
! [A4: real,B3: real] :
( ( ( uminus_uminus_real @ A4 )
= B3 )
= ( ( uminus_uminus_real @ B3 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_1042_equation__minus__iff,axiom,
! [A4: int,B3: int] :
( ( A4
= ( uminus_uminus_int @ B3 ) )
= ( B3
= ( uminus_uminus_int @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_1043_equation__minus__iff,axiom,
! [A4: risk_Free_account,B3: risk_Free_account] :
( ( A4
= ( uminus3377898441596595772ccount @ B3 ) )
= ( B3
= ( uminus3377898441596595772ccount @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_1044_equation__minus__iff,axiom,
! [A4: real,B3: real] :
( ( A4
= ( uminus_uminus_real @ B3 ) )
= ( B3
= ( uminus_uminus_real @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_1045_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1046_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_1047_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1048_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_1049_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_1050_Fun_Obij__uminus,axiom,
bij_betw_int_int @ uminus_uminus_int @ top_top_set_int @ top_top_set_int ).
% Fun.bij_uminus
thf(fact_1051_Fun_Obij__uminus,axiom,
bij_be7177043168045223902ccount @ uminus3377898441596595772ccount @ top_to4387612366039908569ccount @ top_to4387612366039908569ccount ).
% Fun.bij_uminus
thf(fact_1052_Fun_Obij__uminus,axiom,
bij_betw_real_real @ uminus_uminus_real @ top_top_set_real @ top_top_set_real ).
% Fun.bij_uminus
thf(fact_1053_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1054_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_1055_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_1056_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1057_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_1058_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1059_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1060_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1061_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1062_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1063_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1064_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1065_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1066_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1067_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1068_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1069_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N ) )
=> ( P @ N ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1070_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_1071_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1072_inverse__le__iff__le,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_1073_inverse__le__iff__le__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_1074_inverse__less__iff__less,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_real @ B3 @ A4 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_1075_inverse__less__iff__less__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_real @ B3 @ A4 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_1076_inverse__negative__iff__negative,axiom,
! [A4: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ zero_zero_real )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_1077_Rep__account__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N ) ) ) ) ).
% Rep_account_uminus
thf(fact_1078_just__cash__uminus,axiom,
! [A4: real] :
( ( uminus3377898441596595772ccount @ ( risk_Free_just_cash @ A4 ) )
= ( risk_Free_just_cash @ ( uminus_uminus_real @ A4 ) ) ) ).
% just_cash_uminus
thf(fact_1079_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1080_finite__compl,axiom,
! [A: set_option_nat] :
( ( finite5523153139673422903on_nat @ A )
=> ( ( finite5523153139673422903on_nat @ ( uminus2023361477510803743on_nat @ A ) )
= ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat ) ) ) ).
% finite_compl
thf(fact_1081_finite__compl,axiom,
! [A: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ A )
=> ( ( finite6187706683773761046at_nat @ ( uminus4691066839086339966at_nat @ A ) )
= ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat ) ) ) ).
% finite_compl
thf(fact_1082_finite__compl,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_real @ ( uminus612125837232591019t_real @ A ) )
= ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_compl
thf(fact_1083_finite__compl,axiom,
! [A: set_nat_real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( finite7853608736407863218t_real @ ( uminus5090605358382610586t_real @ A ) )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ) ).
% finite_compl
thf(fact_1084_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_1085_inverse__nonzero__iff__nonzero,axiom,
! [A4: real] :
( ( ( inverse_inverse_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1086_inverse__nonpositive__iff__nonpositive,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A4 ) @ zero_zero_real )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_1087_inverse__nonnegative__iff__nonnegative,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A4 ) )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_1088_inverse__positive__iff__positive,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A4 ) )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% inverse_positive_iff_positive
thf(fact_1089_net__asset__value__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( uminus_uminus_real @ ( risk_F2906766666041932210_value @ Alpha2 ) ) ) ).
% net_asset_value_uminus
thf(fact_1090_shortest__period__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F4612863212915232279period @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( risk_F4612863212915232279period @ Alpha2 ) ) ).
% shortest_period_uminus
thf(fact_1091_linordered__field__no__ub,axiom,
! [X8: real] :
? [X_1: real] : ( ord_less_real @ X8 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_1092_linordered__field__no__lb,axiom,
! [X8: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X8 ) ).
% linordered_field_no_lb
thf(fact_1093_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_1094_inverse__zero__imp__zero,axiom,
! [A4: real] :
( ( ( inverse_inverse_real @ A4 )
= zero_zero_real )
=> ( A4 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_1095_nonzero__inverse__eq__imp__eq,axiom,
! [A4: real,B3: real] :
( ( ( inverse_inverse_real @ A4 )
= ( inverse_inverse_real @ B3 ) )
=> ( ( A4 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( A4 = B3 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_1096_nonzero__inverse__inverse__eq,axiom,
! [A4: real] :
( ( A4 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A4 ) )
= A4 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_1097_nonzero__imp__inverse__nonzero,axiom,
! [A4: real] :
( ( A4 != zero_zero_real )
=> ( ( inverse_inverse_real @ A4 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1098_positive__imp__inverse__positive,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A4 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_1099_negative__imp__inverse__negative,axiom,
! [A4: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_1100_inverse__positive__imp__positive,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A4 ) )
=> ( ( A4 != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A4 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_1101_inverse__negative__imp__negative,axiom,
! [A4: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ zero_zero_real )
=> ( ( A4 != zero_zero_real )
=> ( ord_less_real @ A4 @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_1102_less__imp__inverse__less__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A4 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_1103_inverse__less__imp__less__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A4 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_1104_less__imp__inverse__less,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A4 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_1105_inverse__less__imp__less,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_real @ B3 @ A4 ) ) ) ).
% inverse_less_imp_less
thf(fact_1106_nonzero__inverse__minus__eq,axiom,
! [A4: real] :
( ( A4 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A4 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A4 ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_1107_le__imp__inverse__le__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A4 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_1108_inverse__le__imp__le__neg,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A4 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_1109_le__imp__inverse__le,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A4 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_1110_inverse__le__imp__le,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A4 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ord_less_eq_real @ B3 @ A4 ) ) ) ).
% inverse_le_imp_le
thf(fact_1111_Compl__iff,axiom,
! [C: nat > real,A: set_nat_real] :
( ( member_nat_real @ C @ ( uminus5090605358382610586t_real @ A ) )
= ( ~ ( member_nat_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1112_Compl__iff,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1113_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1114_ComplI,axiom,
! [C: nat > real,A: set_nat_real] :
( ~ ( member_nat_real @ C @ A )
=> ( member_nat_real @ C @ ( uminus5090605358382610586t_real @ A ) ) ) ).
% ComplI
thf(fact_1115_ComplI,axiom,
! [C: real,A: set_real] :
( ~ ( member_real @ C @ A )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_1116_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_1117_ComplD,axiom,
! [C: nat > real,A: set_nat_real] :
( ( member_nat_real @ C @ ( uminus5090605358382610586t_real @ A ) )
=> ~ ( member_nat_real @ C @ A ) ) ).
% ComplD
thf(fact_1118_ComplD,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C @ A ) ) ).
% ComplD
thf(fact_1119_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_1120_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1121_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1122_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M6: nat] :
( ( ord_less_nat @ zero_zero_nat @ M6 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1123_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_1124_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
= ( ring_1_of_int_real @ Z ) ) ) ).
% of_nat_nat
thf(fact_1125_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_1126_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_1127_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1128_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1129_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_1130_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1131_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1132_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1133_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1134_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_1135_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_1136_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_1137_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_1138_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_1139_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_1140_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1141_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1142_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_1143_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_1144_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1145_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1146_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: real,B3: real,C: real] :
( ( times_times_real @ ( times_times_real @ A4 @ B3 ) @ C )
= ( times_times_real @ A4 @ ( times_times_real @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1147_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A4 @ B3 ) @ C )
= ( times_times_nat @ A4 @ ( times_times_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1148_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A4: int,B3: int,C: int] :
( ( times_times_int @ ( times_times_int @ A4 @ B3 ) @ C )
= ( times_times_int @ A4 @ ( times_times_int @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1149_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1150_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1151_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1152_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1153_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1154_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1155_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1156_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1157_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1158_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1159_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1160_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1161_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1162_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1163_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1164_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1165_nat__mult__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1166_nat__mult__distrib__neg,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1167_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1168_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1169_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1170_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1171_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1172_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1173_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1174_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1175_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1176_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1177_real__of__int__div4,axiom,
! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1178_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1179_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1180_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1181_verit__less__mono__div__int2,axiom,
! [A: int,B: int,N2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B @ N2 ) @ ( divide_divide_int @ A @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1182_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1183_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1184_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1185_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1186_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1187_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1188_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_1189_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1190_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1191_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1192_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1193_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( divide_divide_int @ A4 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1194_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A4 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1195_div__neg__pos__less0,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1196_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1197_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1198_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N2 )
= ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1199_zdiv__mono1,axiom,
! [A4: int,A7: int,B3: int] :
( ( ord_less_eq_int @ A4 @ A7 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B3 ) @ ( divide_divide_int @ A7 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_1200_zdiv__mono2,axiom,
! [A4: int,B8: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B8 )
=> ( ( ord_less_eq_int @ B8 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B3 ) @ ( divide_divide_int @ A4 @ B8 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1201_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1202_zdiv__mono1__neg,axiom,
! [A4: int,A7: int,B3: int] :
( ( ord_less_eq_int @ A4 @ A7 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B3 ) @ ( divide_divide_int @ A4 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1203_zdiv__mono2__neg,axiom,
! [A4: int,B8: int,B3: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B8 )
=> ( ( ord_less_eq_int @ B8 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B8 ) @ ( divide_divide_int @ A4 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1204_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1205_div__nonneg__neg__le0,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1206_div__nonpos__pos__le0,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1207_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1208_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B3 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1209_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1210_nonneg1__imp__zdiv__pos__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1211_zdiv__zmult2__eq,axiom,
! [C: int,A4: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A4 @ ( times_times_int @ B3 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A4 @ B3 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1212_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1213_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1214_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1215_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W2: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
= ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1216_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N2: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1217_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1218_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_1219_nat__ceiling__le__eq,axiom,
! [X: real,A4: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A4 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A4 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1220_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1221_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1222_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1223_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1224_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1225_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1226_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1227_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1228_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1229_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_1230_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( M = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1231_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1232_div__eq__minus1,axiom,
! [B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1233_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1234_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1235_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1236_just__cash__subtract,axiom,
! [A4: real,B3: real] :
( ( minus_4846202936726426316ccount @ ( risk_Free_just_cash @ A4 ) @ ( risk_Free_just_cash @ B3 ) )
= ( risk_Free_just_cash @ ( minus_minus_real @ A4 @ B3 ) ) ) ).
% just_cash_subtract
thf(fact_1237_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1238_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1239_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1240_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1241_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1242_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1243_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1244_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1245_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1246_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1247_net__asset__value__minus,axiom,
! [Alpha2: risk_Free_account,Beta: risk_Free_account] :
( ( risk_F2906766666041932210_value @ ( minus_4846202936726426316ccount @ Alpha2 @ Beta ) )
= ( minus_minus_real @ ( risk_F2906766666041932210_value @ Alpha2 ) @ ( risk_F2906766666041932210_value @ Beta ) ) ) ).
% net_asset_value_minus
thf(fact_1248_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1249_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1250_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1251_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1252_diff__less__mono,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1253_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1254_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1255_le__diff__iff_H,axiom,
! [A4: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A4 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1256_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1257_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1258_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1259_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1260_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1261_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1262_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1263_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1264_nat__diff__distrib,axiom,
! [Z6: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ord_less_eq_int @ Z6 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1265_int__ops_I6_J,axiom,
! [A4: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A4 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A4 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1266_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha @ Tau ) ) ) ) ) ).
% valid_transfer_def
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_nat @ ( ordina7525502726642723294al_nat @ zero_zero_real @ top_top_set_nat @ ( risk_F170160801229183585ccount @ alpha ) ) ).
%------------------------------------------------------------------------------