TPTP Problem File: SLH0878^1.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_00361_011275__5784542_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1490 ( 574 unt; 216 typ; 0 def)
% Number of atoms : 3978 (1246 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10639 ( 422 ~; 67 |; 298 &;8131 @)
% ( 0 <=>;1721 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 798 ( 798 >; 0 *; 0 +; 0 <<)
% Number of symbols : 178 ( 175 usr; 44 con; 0-3 aty)
% Number of variables : 3505 ( 398 ^;2999 !; 108 ?;3505 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:56:40.818
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
ord_less_nat_real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le3527643927072297637t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
ord_less_eq_nat_real: ( nat > real ) > ( nat > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le2908806416726583473t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
top_top_nat_real_o: ( nat > real ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
top_top_int_o: int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_top_set_nat_real: set_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__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__Int__Oint_J_J,type,
top_to6430115241214627170on_int: set_option_int ).
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__Real__Oreal_J_J,type,
top_to853713521313446370n_real: set_option_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
top_to6856727482967805965nt_nat: set_Pr3448869479623346877nt_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mtf__a_J_J,type,
top_to2684549274079787053_int_a: set_Pr4265292341059111293_int_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_to2179722763343057421at_int: set_Pr7995236796853374141at_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_to4855536200657754381t_real: set_Pr320017278500174781t_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
top_to2612598781856825737_nat_a: set_Pr4193341848836149977_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Int__Oint_J_J,type,
top_to863609200447072703_a_int: set_Pr2444352267426396943_a_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
top_to3353692345378799459_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
top_to199073931035899839a_real: set_Pr4180788328258511119a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to8063371432257647191od_a_a: set_Product_prod_a_a ).
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__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__Int__Oint_J_J,type,
top_top_set_set_int: set_set_int ).
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__Real__Oreal_J_J,type,
top_top_set_set_real: set_set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J_J,type,
top_to8848742569205929409nt_nat: set_Sum_sum_int_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mtf__a_J_J,type,
top_to126475393673508729_int_a: set_Sum_sum_int_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Int__Oint_J_J,type,
top_to4171737849581180865at_int: set_Sum_sum_nat_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_to497527773527737537t_real: set_Sum_sum_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
top_to54524901450547413_nat_a: set_Sum_sum_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Int__Oint_J_J,type,
top_to7528907356895570187_a_int: set_Sum_sum_a_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Real__Oreal_J_J,type,
top_to7642813600076812555a_real: set_Sum_sum_a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
top_to8848906000605539851um_a_a: set_Sum_sum_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Ordinal__Arithmetic_Ofin__support_001t__Real__Oreal_001t__Nat__Onat,type,
ordina1579063754167848977al_nat: real > set_nat > set_nat_real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ofinite_Obalanced_001tf__a,type,
risk_Free_balanced_a: ( a > risk_Free_account ) > real > $o ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: 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__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Typedef_Otype__definition_001t__Risk____Free____Lending__Oaccount_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
type_d8982087200295354172t_real: ( risk_Free_account > nat > real ) > ( ( nat > real ) > risk_Free_account ) > set_nat_real > $o ).
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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Risk____Free____Lending__Oaccount,type,
member5612106785598075018ccount: risk_Free_account > set_Ri1641125681238393385ccount > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060L_062,type,
l: a > risk_Free_account ).
thf(sy_v_c,type,
c: real ).
% Relevant facts (1264)
thf(fact_0_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_1_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_2_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [Uu: a] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_3_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [Uu: a] : zero_z1425366712893667068ccount
@ A )
= zero_z1425366712893667068ccount ) ).
% sum.neutral_const
thf(fact_4_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_5_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_6_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_7_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_8_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_9_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_10_UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% UNIV_I
thf(fact_11_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_12_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_13_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_14_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_15_iso__tuple__UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% iso_tuple_UNIV_I
thf(fact_16_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_17_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_18_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_19_iso__tuple__UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_20_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_21_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_22_local_Obalanced__def,axiom,
( risk_Free_balanced_a
= ( ^ [L: a > risk_Free_account,C2: real] :
( ( groups4655409347963886775ccount @ L @ top_top_set_a )
= ( risk_Free_just_cash @ C2 ) ) ) ) ).
% local.balanced_def
thf(fact_23_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_24_local_Ofinite__Rep__account__ledger,axiom,
! [L2: a > risk_Free_account,A: set_a,N: nat] :
( ( risk_F170160801229183585ccount @ ( groups4655409347963886775ccount @ L2 @ A ) @ N )
= ( groups2740460157737275248a_real
@ ^ [A3: a] : ( risk_F170160801229183585ccount @ ( L2 @ A3 ) @ N )
@ A ) ) ).
% local.finite_Rep_account_ledger
thf(fact_25_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_26_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_27_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_28_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_29_top__set__def,axiom,
( top_top_set_nat_real
= ( collect_nat_real @ top_top_nat_real_o ) ) ).
% top_set_def
thf(fact_30_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_31_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_32_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_33_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_34_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_35_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_36_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_37_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_38_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_39_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_40_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_41_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_42_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_43_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_44_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_58_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_59_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_60_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_61_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_62_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_63_order__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_64_order__less__subst2,axiom,
! [A2: int,B: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_65_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_66_order__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_67_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_68_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_69_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_70_order__less__subst1,axiom,
! [A2: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_71_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_72_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_73_order__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_74_order__less__subst1,axiom,
! [A2: int,F: risk_Free_account > int,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_75_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_76_order__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_77_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_78_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_79_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_80_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_81_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_82_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_83_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_84_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_85_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_86_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_87_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_88_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_89_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_90_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_91_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_92_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_93_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_94_ord__eq__less__subst,axiom,
! [A2: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_95_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_96_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_98_ord__eq__less__subst,axiom,
! [A2: risk_Free_account,F: int > risk_Free_account,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_99_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_100_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_101_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_102_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_103_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_104_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_105_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_106_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_107_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_108_order__less__asym_H,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_109_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_110_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_111_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_112_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_113_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_114_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_115_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_116_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_117_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_118_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_119_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_120_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_121_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_122_dual__order_Ostrict__implies__not__eq,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_123_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_124_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_125_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_126_order_Ostrict__implies__not__eq,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_127_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_128_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_129_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_130_dual__order_Ostrict__trans,axiom,
! [B: risk_Free_account,A2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_131_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_132_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_133_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_134_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_135_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_136_order_Ostrict__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_137_order_Ostrict__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_138_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_139_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B2: int] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_140_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B2: real] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_141_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_142_mem__Collect__eq,axiom,
! [A2: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A2 @ ( collect_nat_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_143_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A: set_nat_real] :
( ( collect_nat_real
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_153_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_154_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_155_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_156_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_157_dual__order_Oirrefl,axiom,
! [A2: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_158_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_159_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_160_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_161_dual__order_Oasym,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
=> ~ ( ord_le2131251472502387783ccount @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_162_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_163_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_164_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_165_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_166_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_167_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_168_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_169_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_170_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_171_ord__less__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_172_ord__less__eq__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2131251472502387783ccount @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_173_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_174_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_175_ord__eq__less__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_176_ord__eq__less__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A2 = B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_177_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_178_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_179_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_180_order_Oasym,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A2 ) ) ).
% order.asym
thf(fact_181_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_182_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_183_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_184_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_185_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_186_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_187_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_188_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_189_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_190_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_191_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_192_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_193_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_194_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_195_UNIV__witness,axiom,
? [X2: nat > real] : ( member_nat_real @ X2 @ top_top_set_nat_real ) ).
% UNIV_witness
thf(fact_196_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_197_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_198_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_199_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_200_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_201_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_202_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_203_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_204_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_205_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_206_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_207_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_208_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_209_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: a > real,J: real > a,T2: set_a,H: a > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_real @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_210_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: a > int,J: int > a,T2: set_a,H: a > real,G: int > real] :
( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_211_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: a > real,J: real > a,T2: set_a,H: a > risk_Free_account,G: real > risk_Free_account] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_real @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S2 )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_212_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: a > int,J: int > a,T2: set_a,H: a > risk_Free_account,G: int > risk_Free_account] :
( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2220918773033463387ccount @ G @ S2 )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_213_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: a > nat,J: nat > a,T2: set_a,H: a > risk_Free_account,G: nat > risk_Free_account] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S2 )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_214_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: nat > real,J: real > nat,T2: set_nat,H: nat > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_real @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_215_sum_Oreindex__bij__witness,axiom,
! [S2: set_int,I: nat > int,J: int > nat,T2: set_nat,H: nat > real,G: int > real] :
( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_int @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_216_sum_Oreindex__bij__witness,axiom,
! [S2: set_a,I: real > a,J: a > real,T2: set_real,H: real > real,G: a > real] :
( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( member_real @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( member_a @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_217_sum_Oreindex__bij__witness,axiom,
! [S2: set_a,I: int > a,J: a > int,T2: set_int,H: int > real,G: a > real] :
( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( member_int @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T2 )
=> ( member_a @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_218_sum_Oreindex__bij__witness,axiom,
! [S2: set_a,I: a > a,J: a > a,T2: set_a,H: a > real,G: a > real] :
( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_a @ ( I @ B2 ) @ S2 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_219_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > real,A: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_220_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > int,A: set_int,H: int > a,Gamma: a > real,Phi: int > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_int @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_221_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > real,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_222_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > int,A: set_int,H: int > a,Gamma: a > risk_Free_account,Phi: int > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_int @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2220918773033463387ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_223_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > nat,A: set_nat,H: nat > a,Gamma: a > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_224_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_225_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > int,A: set_int,H: int > nat,Gamma: nat > real,Phi: int > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_int @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_226_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > a,A: set_a,H: a > real,Gamma: real > real,Phi: a > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_227_sum_Oeq__general__inverses,axiom,
! [B3: set_int,K: int > a,A: set_a,H: a > int,Gamma: int > real,Phi: a > real] :
( ! [Y2: int] :
( ( member_int @ Y2 @ B3 )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_int @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8778361861064173332t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_228_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > a,A: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_a @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_229_sum_Oeq__general,axiom,
! [B3: set_a,A: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_230_sum_Oeq__general,axiom,
! [B3: set_a,A: set_int,H: int > a,Gamma: a > real,Phi: int > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_231_sum_Oeq__general,axiom,
! [B3: set_a,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_232_sum_Oeq__general,axiom,
! [B3: set_a,A: set_int,H: int > a,Gamma: a > risk_Free_account,Phi: int > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2220918773033463387ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_233_sum_Oeq__general,axiom,
! [B3: set_a,A: set_nat,H: nat > a,Gamma: a > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_234_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X5: real] :
( ( member_real @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_235_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_int,H: int > nat,Gamma: nat > real,Phi: int > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X5: int] :
( ( member_int @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_236_sum_Oeq__general,axiom,
! [B3: set_real,A: set_a,H: a > real,Gamma: real > real,Phi: a > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_237_sum_Oeq__general,axiom,
! [B3: set_int,A: set_a,H: a > int,Gamma: int > real,Phi: a > real] :
( ! [Y2: int] :
( ( member_int @ Y2 @ B3 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_int @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups8778361861064173332t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_238_sum_Oeq__general,axiom,
! [B3: set_a,A: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_239_sum_Ocong,axiom,
! [A: set_a,B3: set_a,G: a > real,H: a > real] :
( ( A = B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ A )
= ( groups2740460157737275248a_real @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_240_sum_Ocong,axiom,
! [A: set_a,B3: set_a,G: a > risk_Free_account,H: a > risk_Free_account] :
( ( A = B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4655409347963886775ccount @ G @ A )
= ( groups4655409347963886775ccount @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_241_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G: nat > real,H: nat > real] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_242_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X4: a] : $true ) ) ).
% UNIV_def
thf(fact_243_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X4: nat] : $true ) ) ).
% UNIV_def
thf(fact_244_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X4: int] : $true ) ) ).
% UNIV_def
thf(fact_245_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X4: real] : $true ) ) ).
% UNIV_def
thf(fact_246_UNIV__def,axiom,
( top_top_set_nat_real
= ( collect_nat_real
@ ^ [X4: nat > real] : $true ) ) ).
% UNIV_def
thf(fact_247_sum_Oswap,axiom,
! [G: a > a > real,B3: set_a,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B3 )
@ A )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_248_sum_Oswap,axiom,
! [G: a > nat > real,B3: set_nat,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B3 )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_249_sum_Oswap,axiom,
! [G: a > a > risk_Free_account,B3: set_a,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [I2: a] : ( groups4655409347963886775ccount @ ( G @ I2 ) @ B3 )
@ A )
= ( groups4655409347963886775ccount
@ ^ [J2: a] :
( groups4655409347963886775ccount
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_250_sum_Oswap,axiom,
! [G: nat > a > real,B3: set_a,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B3 )
@ A )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_251_sum_Oswap,axiom,
! [G: nat > nat > real,B3: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B3 )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_252_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_253_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_254_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_255_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_256_top_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != top_top_set_a )
= ( ord_less_set_a @ A2 @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_257_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_258_top_Onot__eq__extremum,axiom,
! [A2: set_int] :
( ( A2 != top_top_set_int )
= ( ord_less_set_int @ A2 @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_259_top_Onot__eq__extremum,axiom,
! [A2: set_real] :
( ( A2 != top_top_set_real )
= ( ord_less_set_real @ A2 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_260_top_Onot__eq__extremum,axiom,
! [A2: set_nat_real] :
( ( A2 != top_top_set_nat_real )
= ( ord_le3527643927072297637t_real @ A2 @ top_top_set_nat_real ) ) ).
% top.not_eq_extremum
thf(fact_261_top_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A2 ) ).
% top.extremum_strict
thf(fact_262_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_263_top_Oextremum__strict,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A2 ) ).
% top.extremum_strict
thf(fact_264_top_Oextremum__strict,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A2 ) ).
% top.extremum_strict
thf(fact_265_top_Oextremum__strict,axiom,
! [A2: set_nat_real] :
~ ( ord_le3527643927072297637t_real @ top_top_set_nat_real @ A2 ) ).
% top.extremum_strict
thf(fact_266_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_267_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_268_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_269_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_270_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_271_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_272_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_273_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > risk_Free_account,A: set_int] :
( ( ( groups2220918773033463387ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_274_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > risk_Free_account,A: set_nat] :
( ( ( groups6033208628184776703ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_275_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_276_sum_Oneutral,axiom,
! [A: set_a,G: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups2740460157737275248a_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_277_sum_Oneutral,axiom,
! [A: set_a,G: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups4655409347963886775ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.neutral
thf(fact_278_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_279_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_280_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_281_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_282_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_283_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_284_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C2: real] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_285_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_286_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_287_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_288_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_289_local_Ofinite__UNIV,axiom,
finite_finite_a @ top_top_set_a ).
% local.finite_UNIV
thf(fact_290_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_291_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_292_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_293_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_294_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_295_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_296_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_297_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_298_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_299_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X4: int] : ( member_int @ X4 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_300_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X4: real] : ( member_real @ X4 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_301_top__empty__eq,axiom,
( top_top_nat_real_o
= ( ^ [X4: nat > real] : ( member_nat_real @ X4 @ top_top_set_nat_real ) ) ) ).
% top_empty_eq
thf(fact_302_local_Ofinite,axiom,
! [A: set_a] : ( finite_finite_a @ A ) ).
% local.finite
thf(fact_303_of__nat__eq__iff,axiom,
! [M3: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M3 = N ) ) ).
% of_nat_eq_iff
thf(fact_304_of__nat__eq__iff,axiom,
! [M3: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M3 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M3 = N ) ) ).
% of_nat_eq_iff
thf(fact_305_local_Ofinite__code,axiom,
( finite_finite_a
= ( ^ [A5: set_a] : $true ) ) ).
% local.finite_code
thf(fact_306_sum_Oinfinite,axiom,
! [A: set_a,G: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_307_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_308_sum_Oinfinite,axiom,
! [A: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_309_sum_Oinfinite,axiom,
! [A: set_int,G: int > real] :
( ~ ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_310_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > risk_Free_account] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_311_sum_Oinfinite,axiom,
! [A: set_int,G: int > risk_Free_account] :
( ~ ( finite_finite_int @ A )
=> ( ( groups2220918773033463387ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_312_sum_Oinfinite,axiom,
! [A: set_a,G: a > int] :
( ~ ( finite_finite_a @ A )
=> ( ( groups6332066207828071664_a_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_313_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_314_sum_Oinfinite,axiom,
! [A: set_int,G: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_315_sum_Oinfinite,axiom,
! [A: set_a,G: a > real] :
( ~ ( finite_finite_a @ A )
=> ( ( groups2740460157737275248a_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_316_sum__eq__0__iff,axiom,
! [F2: set_a,F: a > nat] :
( ( finite_finite_a @ F2 )
=> ( ( ( groups6334556678337121940_a_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: a] :
( ( member_a @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_317_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_318_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: int] :
( ( member_int @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_319_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_320_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_321_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_322_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_323_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_324_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_325_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_326_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_327_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri5074537144036343181t_real @ M3 )
= zero_zero_real )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_328_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_329_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_330_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_331_sum_Odelta,axiom,
! [S2: set_real,A2: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K2: real] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K2: real] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_332_sum_Odelta,axiom,
! [S2: set_a,A2: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A2 @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_333_sum_Odelta,axiom,
! [S2: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_334_sum_Odelta,axiom,
! [S2: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_335_sum_Odelta,axiom,
! [S2: set_real,A2: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K2: real] : ( if_real @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K2: real] : ( if_real @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_336_sum_Odelta,axiom,
! [S2: set_int,A2: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_337_sum_Odelta,axiom,
! [S2: set_real,A2: real,B: real > risk_Free_account] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups8516999891779824987ccount
@ ^ [K2: real] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups8516999891779824987ccount
@ ^ [K2: real] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_338_sum_Odelta,axiom,
! [S2: set_nat,A2: nat,B: nat > risk_Free_account] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K2: nat] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K2: nat] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_339_sum_Odelta,axiom,
! [S2: set_int,A2: int,B: int > risk_Free_account] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups2220918773033463387ccount
@ ^ [K2: int] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups2220918773033463387ccount
@ ^ [K2: int] : ( if_Risk_Free_account @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_340_sum_Odelta,axiom,
! [S2: set_real,A2: real,B: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K2: real] : ( if_int @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K2: real] : ( if_int @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_341_sum_Odelta_H,axiom,
! [S2: set_real,A2: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K2: real] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K2: real] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_342_sum_Odelta_H,axiom,
! [S2: set_a,A2: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A2 @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_343_sum_Odelta_H,axiom,
! [S2: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_344_sum_Odelta_H,axiom,
! [S2: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_345_sum_Odelta_H,axiom,
! [S2: set_real,A2: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K2: real] : ( if_real @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K2: real] : ( if_real @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_346_sum_Odelta_H,axiom,
! [S2: set_int,A2: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_347_sum_Odelta_H,axiom,
! [S2: set_real,A2: real,B: real > risk_Free_account] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups8516999891779824987ccount
@ ^ [K2: real] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups8516999891779824987ccount
@ ^ [K2: real] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_348_sum_Odelta_H,axiom,
! [S2: set_nat,A2: nat,B: nat > risk_Free_account] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K2: nat] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K2: nat] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_349_sum_Odelta_H,axiom,
! [S2: set_int,A2: int,B: int > risk_Free_account] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A2 @ S2 )
=> ( ( groups2220918773033463387ccount
@ ^ [K2: int] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S2 )
=> ( ( groups2220918773033463387ccount
@ ^ [K2: int] : ( if_Risk_Free_account @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_z1425366712893667068ccount )
@ S2 )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_350_sum_Odelta_H,axiom,
! [S2: set_real,A2: real,B: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A2 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K2: real] : ( if_int @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_real @ A2 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K2: real] : ( if_int @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_351_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_352_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_353_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_354_sum_Oswap__restrict,axiom,
! [A: set_real,B3: set_a,G: real > a > real,R: real > a > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups8097168146408367636l_real
@ ^ [X4: real] :
( groups2740460157737275248a_real @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups8097168146408367636l_real
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_355_sum_Oswap__restrict,axiom,
! [A: set_int,B3: set_a,G: int > a > real,R: int > a > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups8778361861064173332t_real
@ ^ [X4: int] :
( groups2740460157737275248a_real @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups8778361861064173332t_real
@ ^ [X4: int] : ( G @ X4 @ Y4 )
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_356_sum_Oswap__restrict,axiom,
! [A: set_real,B3: set_a,G: real > a > risk_Free_account,R: real > a > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups8516999891779824987ccount
@ ^ [X4: real] :
( groups4655409347963886775ccount @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups8516999891779824987ccount
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_357_sum_Oswap__restrict,axiom,
! [A: set_nat,B3: set_a,G: nat > a > risk_Free_account,R: nat > a > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups6033208628184776703ccount
@ ^ [X4: nat] :
( groups4655409347963886775ccount @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups6033208628184776703ccount
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_358_sum_Oswap__restrict,axiom,
! [A: set_int,B3: set_a,G: int > a > risk_Free_account,R: int > a > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups2220918773033463387ccount
@ ^ [X4: int] :
( groups4655409347963886775ccount @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups2220918773033463387ccount
@ ^ [X4: int] : ( G @ X4 @ Y4 )
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_359_sum_Oswap__restrict,axiom,
! [A: set_real,B3: set_nat,G: real > nat > real,R: real > nat > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups8097168146408367636l_real
@ ^ [X4: real] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8097168146408367636l_real
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_360_sum_Oswap__restrict,axiom,
! [A: set_int,B3: set_nat,G: int > nat > real,R: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups8778361861064173332t_real
@ ^ [X4: int] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8778361861064173332t_real
@ ^ [X4: int] : ( G @ X4 @ Y4 )
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_361_sum_Oswap__restrict,axiom,
! [A: set_a,B3: set_real,G: a > real > real,R: a > real > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_real @ B3 )
=> ( ( groups2740460157737275248a_real
@ ^ [X4: a] :
( groups8097168146408367636l_real @ ( G @ X4 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8097168146408367636l_real
@ ^ [Y4: real] :
( groups2740460157737275248a_real
@ ^ [X4: a] : ( G @ X4 @ Y4 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_362_sum_Oswap__restrict,axiom,
! [A: set_a,B3: set_int,G: a > int > real,R: a > int > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ( groups2740460157737275248a_real
@ ^ [X4: a] :
( groups8778361861064173332t_real @ ( G @ X4 )
@ ( collect_int
@ ^ [Y4: int] :
( ( member_int @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8778361861064173332t_real
@ ^ [Y4: int] :
( groups2740460157737275248a_real
@ ^ [X4: a] : ( G @ X4 @ Y4 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_363_sum_Oswap__restrict,axiom,
! [A: set_a,B3: set_a,G: a > a > real,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ( groups2740460157737275248a_real
@ ^ [X4: a] :
( groups2740460157737275248a_real @ ( G @ X4 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B3 )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups2740460157737275248a_real
@ ^ [X4: a] : ( G @ X4 @ Y4 )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_364_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_365_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_366_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_367_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_368_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_369_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_370_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_371_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_372_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_373_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_374_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_375_sum_Ointer__filter,axiom,
! [A: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X4: real] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_376_sum_Ointer__filter,axiom,
! [A: set_a,G: a > nat,P: a > $o] :
( ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups6334556678337121940_a_nat
@ ^ [X4: a] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_377_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_378_sum_Ointer__filter,axiom,
! [A: set_int,G: int > nat,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_379_sum_Ointer__filter,axiom,
! [A: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_380_sum_Ointer__filter,axiom,
! [A: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X4: int] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_381_sum_Ointer__filter,axiom,
! [A: set_real,G: real > risk_Free_account,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8516999891779824987ccount @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8516999891779824987ccount
@ ^ [X4: real] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_382_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > risk_Free_account,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups6033208628184776703ccount
@ ^ [X4: nat] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_383_sum_Ointer__filter,axiom,
! [A: set_int,G: int > risk_Free_account,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups2220918773033463387ccount @ G
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups2220918773033463387ccount
@ ^ [X4: int] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_384_sum_Ointer__filter,axiom,
! [A: set_real,G: real > int,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1932886352136224148al_int @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups1932886352136224148al_int
@ ^ [X4: real] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_385_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_386_of__nat__sum,axiom,
! [F: a > nat,A: set_a] :
( ( semiri5074537144036343181t_real @ ( groups6334556678337121940_a_nat @ F @ A ) )
= ( groups2740460157737275248a_real
@ ^ [X4: a] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_387_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_388_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
~ ( P @ X4 ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_389_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_390_finite__Collect__not,axiom,
! [P: int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
~ ( P @ X4 ) ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Collect_not
thf(fact_391_finite__Collect__not,axiom,
! [P: real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
~ ( P @ X4 ) ) )
= ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_Collect_not
thf(fact_392_finite__Collect__not,axiom,
! [P: ( nat > real ) > $o] :
( ( finite7853608736407863218t_real @ ( collect_nat_real @ P ) )
=> ( ( finite7853608736407863218t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
~ ( P @ X4 ) ) )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ) ).
% finite_Collect_not
thf(fact_393_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_394_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_395_finite__option__UNIV,axiom,
( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% finite_option_UNIV
thf(fact_396_finite__option__UNIV,axiom,
( ( finite5731380839103853331n_real @ top_to853713521313446370n_real )
= ( finite_finite_real @ top_top_set_real ) ) ).
% finite_option_UNIV
thf(fact_397_finite__option__UNIV,axiom,
( ( finite4829245213595033346t_real @ top_to2669572386385008721t_real )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ).
% finite_option_UNIV
thf(fact_398_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_399_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_400_finite__Plus__UNIV__iff,axiom,
( ( finite5547626034989006084_a_int @ top_to7528907356895570187_a_int )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_401_finite__Plus__UNIV__iff,axiom,
( ( finite5047159645595456260a_real @ top_to7642813600076812555a_real )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_402_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_403_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_404_finite__Plus__UNIV__iff,axiom,
( ( finite2009855664264564338at_int @ top_to4171737849581180865at_int )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_405_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_406_finite__Plus__UNIV__iff,axiom,
( ( finite1495123513696658034_int_a @ top_to126475393673508729_int_a )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_407_finite__Plus__UNIV__iff,axiom,
( ( finite7187060395674815602nt_nat @ top_to8848742569205929409nt_nat )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_408_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_409_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_410_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_411_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_412_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_413_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_414_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_415_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_416_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_417_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_418_Finite__Set_Ofinite__set,axiom,
( ( finite6197958912794628473et_int @ top_top_set_set_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% Finite_Set.finite_set
thf(fact_419_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_420_Finite__Set_Ofinite__set,axiom,
( ( finite7096078154069415912t_real @ top_to1863808837862421559t_real )
= ( finite7853608736407863218t_real @ top_top_set_nat_real ) ) ).
% Finite_Set.finite_set
thf(fact_421_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_422_finite__interval__int4,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_423_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_424_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_425_int__int__eq,axiom,
! [M3: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M3 = N ) ) ).
% int_int_eq
thf(fact_426_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_427_finite__psubset__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B4: set_a] :
( ( ord_less_set_a @ B4 @ A6 )
=> ( P @ B4 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_428_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A6 )
=> ( P @ B4 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_429_finite__psubset__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [B4: set_int] :
( ( ord_less_set_int @ B4 @ A6 )
=> ( P @ B4 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_430_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_431_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_432_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_433_pigeonhole__infinite__rel,axiom,
! [A: set_real,B3: set_a,R: real > a > $o] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_434_pigeonhole__infinite__rel,axiom,
! [A: set_real,B3: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_435_pigeonhole__infinite__rel,axiom,
! [A: set_real,B3: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_436_pigeonhole__infinite__rel,axiom,
! [A: set_a,B3: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_437_pigeonhole__infinite__rel,axiom,
! [A: set_a,B3: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_438_pigeonhole__infinite__rel,axiom,
! [A: set_a,B3: set_int,R: a > int > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_439_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_440_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_441_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_442_pigeonhole__infinite__rel,axiom,
! [A: set_int,B3: set_a,R: int > a > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_443_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_444_infinite__UNIV__char__0,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_char_0
thf(fact_445_infinite__UNIV__char__0,axiom,
~ ( finite_finite_real @ top_top_set_real ) ).
% infinite_UNIV_char_0
thf(fact_446_ex__new__if__finite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A )
=> ? [A4: a] :
~ ( member_a @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_447_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_448_ex__new__if__finite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_int @ A )
=> ? [A4: int] :
~ ( member_int @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_449_ex__new__if__finite,axiom,
! [A: set_real] :
( ~ ( finite_finite_real @ top_top_set_real )
=> ( ( finite_finite_real @ A )
=> ? [A4: real] :
~ ( member_real @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_450_ex__new__if__finite,axiom,
! [A: set_nat_real] :
( ~ ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ( ( finite7853608736407863218t_real @ A )
=> ? [A4: nat > real] :
~ ( member_nat_real @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_451_finite__fun__UNIVD2,axiom,
( ( finite7853608736407863218t_real @ top_top_set_nat_real )
=> ( finite_finite_real @ top_top_set_real ) ) ).
% finite_fun_UNIVD2
thf(fact_452_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_453_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_454_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite2467047343636934000_a_int @ top_to863609200447072703_a_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_455_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_real @ top_top_set_real )
=> ( finite4642337302493810032a_real @ top_to199073931035899839a_real ) ) ) ).
% finite_Prod_UNIV
thf(fact_456_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_457_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_458_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite1999359929226648326at_int @ top_to2179722763343057421at_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_459_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_460_finite__Prod__UNIV,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite7637916859199361758_int_a @ top_to2684549274079787053_int_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_461_finite__Prod__UNIV,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_462_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_463_finite__prod,axiom,
( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_464_finite__prod,axiom,
( ( finite2467047343636934000_a_int @ top_to863609200447072703_a_int )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_465_finite__prod,axiom,
( ( finite4642337302493810032a_real @ top_to199073931035899839a_real )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_real @ top_top_set_real ) ) ) ).
% finite_prod
thf(fact_466_finite__prod,axiom,
( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_467_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_468_finite__prod,axiom,
( ( finite1999359929226648326at_int @ top_to2179722763343057421at_int )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_469_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_470_finite__prod,axiom,
( ( finite7637916859199361758_int_a @ top_to2684549274079787053_int_a )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_471_finite__prod,axiom,
( ( finite7176564660636899590nt_nat @ top_to6856727482967805965nt_nat )
= ( ( finite_finite_int @ top_top_set_int )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_472_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_473_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_474_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_475_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_476_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_477_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_478_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_479_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_480_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_481_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_482_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_483_add_Oinverse__inverse,axiom,
! [A2: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_484_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_485_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_486_neg__equal__iff__equal,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A2 )
= ( uminus3377898441596595772ccount @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_487_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_488_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_489_verit__minus__simplify_I4_J,axiom,
! [B: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_490_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_491_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_492_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_493_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_494_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_495_neg__0__equal__iff__equal,axiom,
! [A2: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A2 ) )
= ( zero_z1425366712893667068ccount = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_496_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_497_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_498_neg__equal__0__iff__equal,axiom,
! [A2: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A2 )
= zero_z1425366712893667068ccount )
= ( A2 = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_499_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_500_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_501_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_502_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_503_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_504_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_505_neg__less__iff__less,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B ) @ ( uminus3377898441596595772ccount @ A2 ) )
= ( ord_le2131251472502387783ccount @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_506_neg__less__iff__less,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_507_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_508_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_509_less__neg__neg,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_510_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_511_neg__less__pos,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_pos
thf(fact_512_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_513_neg__0__less__iff__less,axiom,
! [A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A2 ) )
= ( ord_le2131251472502387783ccount @ A2 @ zero_z1425366712893667068ccount ) ) ).
% neg_0_less_iff_less
thf(fact_514_neg__0__less__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_515_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_516_neg__less__0__iff__less,axiom,
! [A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A2 ) @ zero_z1425366712893667068ccount )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_517_neg__less__0__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_518_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_519_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_520_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_521_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_522_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_523_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_524_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_525_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_526_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_527_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_minus
thf(fact_528_negative__eq__positive,axiom,
! [N: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_529_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_530_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_531_of__int__sum,axiom,
! [F: a > int,A: set_a] :
( ( ring_1_of_int_real @ ( groups6332066207828071664_a_int @ F @ A ) )
= ( groups2740460157737275248a_real
@ ^ [X4: a] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A ) ) ).
% of_int_sum
thf(fact_532_of__int__sum,axiom,
! [F: nat > int,A: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A ) ) ).
% of_int_sum
thf(fact_533_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_534_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_535_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_536_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_537_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_538_ex__less__of__int,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_less_of_int
thf(fact_539_ex__of__int__less,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).
% ex_of_int_less
thf(fact_540_less__set__def,axiom,
( ord_le3527643927072297637t_real
= ( ^ [A5: set_nat_real,B6: set_nat_real] :
( ord_less_nat_real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A5 )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_541_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B6: set_real] :
( ord_less_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_542_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X4: int] : ( member_int @ X4 @ A5 )
@ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_543_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_544_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_545_psubsetD,axiom,
! [A: set_nat_real,B3: set_nat_real,C: nat > real] :
( ( ord_le3527643927072297637t_real @ A @ B3 )
=> ( ( member_nat_real @ C @ A )
=> ( member_nat_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_546_psubsetD,axiom,
! [A: set_real,B3: set_real,C: real] :
( ( ord_less_set_real @ A @ B3 )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_547_psubsetD,axiom,
! [A: set_int,B3: set_int,C: int] :
( ( ord_less_set_int @ A @ B3 )
=> ( ( member_int @ C @ A )
=> ( member_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_548_psubsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_549_psubsetD,axiom,
! [A: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A @ B3 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_550_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_551_verit__negate__coefficient_I2_J,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B ) @ ( uminus3377898441596595772ccount @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_552_verit__negate__coefficient_I2_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_553_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_554_equation__minus__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( A2
= ( uminus3377898441596595772ccount @ B ) )
= ( B
= ( uminus3377898441596595772ccount @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_555_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_556_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_557_minus__equation__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A2 )
= B )
= ( ( uminus3377898441596595772ccount @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_558_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_559_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B: int] :
( ( A2 = B )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_560_verit__negate__coefficient_I3_J,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( A2 = B )
=> ( ( uminus3377898441596595772ccount @ A2 )
= ( uminus3377898441596595772ccount @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_561_verit__negate__coefficient_I3_J,axiom,
! [A2: real,B: real] :
( ( A2 = B )
=> ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_562_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_563_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_564_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_565_less__minus__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ ( uminus3377898441596595772ccount @ B ) )
= ( ord_le2131251472502387783ccount @ B @ ( uminus3377898441596595772ccount @ A2 ) ) ) ).
% less_minus_iff
thf(fact_566_less__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% less_minus_iff
thf(fact_567_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_568_minus__less__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A2 ) @ B )
= ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_569_minus__less__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_570_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_571_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_572_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_573_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_574_sum__negf,axiom,
! [F: a > real,A: set_a] :
( ( groups2740460157737275248a_real
@ ^ [X4: a] : ( uminus_uminus_real @ ( F @ X4 ) )
@ A )
= ( uminus_uminus_real @ ( groups2740460157737275248a_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_575_sum__negf,axiom,
! [F: a > risk_Free_account,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [X4: a] : ( uminus3377898441596595772ccount @ ( F @ X4 ) )
@ A )
= ( uminus3377898441596595772ccount @ ( groups4655409347963886775ccount @ F @ A ) ) ) ).
% sum_negf
thf(fact_576_sum__negf,axiom,
! [F: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( uminus_uminus_real @ ( F @ X4 ) )
@ A )
= ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_577_not__int__zless__negative,axiom,
! [N: nat,M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% not_int_zless_negative
thf(fact_578_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_579_zless__nat__eq__int__zless,axiom,
! [M3: nat,Z: int] :
( ( ord_less_nat @ M3 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_580_int__cases4,axiom,
! [M3: int] :
( ! [N3: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_581_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_582_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_583_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_584_verit__comp__simplify1_I1_J,axiom,
! [A2: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_585_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_586_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_587_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A3: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_588_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_589_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_590_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_591_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_592_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_593_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_594_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_595_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 )
=> ( ( ? [X6: nat > real] : ( F2 @ X6 ) )
= ( ? [B5: risk_Free_account] : ( F2 @ ( Rep @ B5 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_596_Rep__account__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( ^ [N2: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N2 ) ) ) ) ).
% Rep_account_uminus
thf(fact_597_finite__compl,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( uminus_uminus_set_a @ A ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_compl
thf(fact_598_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_599_finite__compl,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ ( uminus1532241313380277803et_int @ A ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_compl
thf(fact_600_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_601_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_602_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N2 ) ) ) ) ) ).
% uminus_account_def
thf(fact_603_nat__less__iff,axiom,
! [W: int,M3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M3 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ).
% nat_less_iff
thf(fact_604_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_605_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_606_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_607_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_608_nat__eq__iff,axiom,
! [W: int,M3: nat] :
( ( ( nat2 @ W )
= M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_609_nat__eq__iff2,axiom,
! [M3: nat,W: int] :
( ( M3
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_610_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_611_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_612_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_613_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_614_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_615_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_616_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_617_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_618_dual__order_Orefl,axiom,
! [A2: risk_Free_account] : ( ord_le4245800335709223507ccount @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_619_ComplI,axiom,
! [C: nat > real,A: set_nat_real] :
( ~ ( member_nat_real @ C @ A )
=> ( member_nat_real @ C @ ( uminus5090605358382610586t_real @ A ) ) ) ).
% ComplI
thf(fact_620_ComplI,axiom,
! [C: real,A: set_real] :
( ~ ( member_real @ C @ A )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_621_ComplI,axiom,
! [C: int,A: set_int] :
( ~ ( member_int @ C @ A )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% ComplI
thf(fact_622_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_623_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_624_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_625_Compl__iff,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_626_Compl__iff,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
= ( ~ ( member_int @ C @ A ) ) ) ).
% Compl_iff
thf(fact_627_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_628_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_629_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_630_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_631_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_632_neg__le__iff__le,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B ) @ ( uminus3377898441596595772ccount @ A2 ) )
= ( ord_le4245800335709223507ccount @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_633_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_634_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_635_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_636_ceiling__of__int,axiom,
! [Z: int] :
( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z ) )
= Z ) ).
% ceiling_of_int
thf(fact_637_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int_real @ N2 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_638_finite__interval__int1,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_639_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_640_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_641_neg__0__le__iff__le,axiom,
! [A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A2 ) )
= ( ord_le4245800335709223507ccount @ A2 @ zero_z1425366712893667068ccount ) ) ).
% neg_0_le_iff_le
thf(fact_642_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_643_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_644_neg__le__0__iff__le,axiom,
! [A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A2 ) @ zero_z1425366712893667068ccount )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_645_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_646_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_647_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_648_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_649_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_650_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_651_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_652_negative__zle,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zle
thf(fact_653_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_654_finite__interval__int3,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_655_finite__interval__int2,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_656_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_657_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_658_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M3 ) @ zero_zero_real )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_659_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_660_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_661_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_662_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_663_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_664_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_665_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_666_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_667_Compl__eq,axiom,
( uminus5090605358382610586t_real
= ( ^ [A5: set_nat_real] :
( collect_nat_real
@ ^ [X4: nat > real] :
~ ( member_nat_real @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_668_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A5: set_real] :
( collect_real
@ ^ [X4: real] :
~ ( member_real @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_669_Compl__eq,axiom,
( uminus_uminus_set_a
= ( ^ [A5: set_a] :
( collect_a
@ ^ [X4: a] :
~ ( member_a @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_670_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A5: set_nat] :
( collect_nat
@ ^ [X4: nat] :
~ ( member_nat @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_671_Compl__eq,axiom,
( uminus1532241313380277803et_int
= ( ^ [A5: set_int] :
( collect_int
@ ^ [X4: int] :
~ ( member_int @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_672_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_673_Collect__neg__eq,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X4: int] :
~ ( P @ X4 ) )
= ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% Collect_neg_eq
thf(fact_674_ComplD,axiom,
! [C: nat > real,A: set_nat_real] :
( ( member_nat_real @ C @ ( uminus5090605358382610586t_real @ A ) )
=> ~ ( member_nat_real @ C @ A ) ) ).
% ComplD
thf(fact_675_ComplD,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C @ A ) ) ).
% ComplD
thf(fact_676_ComplD,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
=> ~ ( member_int @ C @ A ) ) ).
% ComplD
thf(fact_677_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_678_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_679_uminus__set__def,axiom,
( uminus5090605358382610586t_real
= ( ^ [A5: set_nat_real] :
( collect_nat_real
@ ( uminus8324563361911858795real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_680_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A5: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_681_uminus__set__def,axiom,
( uminus_uminus_set_a
= ( ^ [A5: set_a] :
( collect_a
@ ( uminus_uminus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_682_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A5: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_683_uminus__set__def,axiom,
( uminus1532241313380277803et_int
= ( ^ [A5: set_int] :
( collect_int
@ ( uminus_uminus_int_o
@ ^ [X4: int] : ( member_int @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_684_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_685_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_686_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_687_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_688_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_689_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_690_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_691_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_692_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_693_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_694_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_695_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_696_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: risk_Free_account,Z3: risk_Free_account] : ( Y5 = Z3 ) )
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_697_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_698_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_699_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_700_ord__eq__le__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A2 = B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_701_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_702_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_703_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_704_ord__le__eq__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ( B = C )
=> ( ord_le4245800335709223507ccount @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_705_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_706_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_707_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_708_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_709_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_710_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_711_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_712_order_Otrans,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A2 @ C ) ) ) ).
% order.trans
thf(fact_713_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_714_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_715_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_716_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_717_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: int,B2: int] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_718_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_719_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: real,B2: real] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_720_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A3: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A3 )
& ( ord_less_eq_int @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_721_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_722_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A3 )
& ( ord_less_eq_real @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_723_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: risk_Free_account,Z3: risk_Free_account] : ( Y5 = Z3 ) )
= ( ^ [A3: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B5 @ A3 )
& ( ord_le4245800335709223507ccount @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_724_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_725_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_726_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_727_dual__order_Oantisym,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
=> ( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_728_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_729_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_730_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_731_dual__order_Otrans,axiom,
! [B: risk_Free_account,A2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le4245800335709223507ccount @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_732_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_733_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_734_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_735_antisym,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_736_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
& ( ord_less_eq_int @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_737_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_738_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
& ( ord_less_eq_real @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_739_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: risk_Free_account,Z3: risk_Free_account] : ( Y5 = Z3 ) )
= ( ^ [A3: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B5 )
& ( ord_le4245800335709223507ccount @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_740_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_741_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_742_order__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_743_order__subst1,axiom,
! [A2: int,F: risk_Free_account > int,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_744_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_745_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_746_order__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_747_order__subst1,axiom,
! [A2: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_748_order__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_749_order__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_750_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_751_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_752_order__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_753_order__subst2,axiom,
! [A2: int,B: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_754_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_755_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_756_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_757_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_758_order__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_759_order__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_760_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_761_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_762_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_763_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_764_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_765_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_766_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_767_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_768_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_769_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_770_ord__eq__le__subst,axiom,
! [A2: risk_Free_account,F: int > risk_Free_account,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_771_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_772_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_773_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_774_ord__eq__le__subst,axiom,
! [A2: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_775_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_776_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_777_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_778_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_779_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_780_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_781_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_782_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_783_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_784_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_785_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_786_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_ceiling__le,axiom,
! [X: real,A2: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A2 ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A2 ) ) ).
% ceiling_le
thf(fact_795_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_796_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_797_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_798_verit__comp__simplify1_I2_J,axiom,
! [A2: risk_Free_account] : ( ord_le4245800335709223507ccount @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_799_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_800_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_801_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_802_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_803_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_804_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_805_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_806_of__nat__ceiling,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% of_nat_ceiling
thf(fact_807_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_808_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_809_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_810_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_811_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_812_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_813_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
= ( ord_less_real @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_814_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_815_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_816_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_817_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_818_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_819_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_820_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_821_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_822_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_823_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_824_nless__le,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A2 @ B ) )
= ( ~ ( ord_le4245800335709223507ccount @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_825_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_826_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_827_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_828_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_829_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_830_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_831_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_832_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_833_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_834_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_835_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_836_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_837_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_838_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ~ ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_839_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_840_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_841_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_842_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_843_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_844_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_845_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B5: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_846_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_847_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_848_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_849_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_850_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_851_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_852_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_853_order_Ostrict__trans1,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_854_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_855_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_856_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_857_order_Ostrict__trans2,axiom,
! [A2: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_858_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_859_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_860_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_861_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B5 )
& ~ ( ord_le4245800335709223507ccount @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_862_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_863_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_864_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A3: int] :
( ( ord_less_int @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_865_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_866_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A3: real] :
( ( ord_less_real @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_867_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B5: risk_Free_account,A3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_868_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A3: int] :
( ( ord_less_eq_int @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_869_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_870_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A3: real] :
( ( ord_less_eq_real @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_871_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B5: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_872_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_873_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_874_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_875_dual__order_Ostrict__trans1,axiom,
! [B: risk_Free_account,A2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_876_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_877_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_878_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_879_dual__order_Ostrict__trans2,axiom,
! [B: risk_Free_account,A2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_880_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A3: int] :
( ( ord_less_eq_int @ B5 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_881_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_882_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A3: real] :
( ( ord_less_eq_real @ B5 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_883_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B5: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B5 @ A3 )
& ~ ( ord_le4245800335709223507ccount @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_884_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_885_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_886_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_887_order_Ostrict__implies__order,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ord_le4245800335709223507ccount @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_888_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_889_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_890_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_891_dual__order_Ostrict__implies__order,axiom,
! [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
=> ( ord_le4245800335709223507ccount @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_892_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_893_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_894_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_895_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_896_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_897_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_898_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_899_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_900_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_901_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_902_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_903_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_904_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_905_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_906_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_907_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_908_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_909_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_910_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_911_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_912_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_913_order__le__neq__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le2131251472502387783ccount @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_914_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_915_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_916_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_917_order__neq__le__trans,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( A2 != B )
=> ( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ord_le2131251472502387783ccount @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_927_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_928_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_929_order__le__less__subst1,axiom,
! [A2: int,F: risk_Free_account > int,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_930_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_931_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_932_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_933_order__le__less__subst1,axiom,
! [A2: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_934_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_935_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_936_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_937_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_938_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_939_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_940_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_941_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_942_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_943_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_944_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_945_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_946_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_947_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_948_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_949_order__less__le__subst1,axiom,
! [A2: risk_Free_account,F: int > risk_Free_account,B: int,C: int] :
( ( ord_le2131251472502387783ccount @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_950_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_951_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_952_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_953_order__less__le__subst1,axiom,
! [A2: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_954_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_955_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_956_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_957_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_958_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_959_order__less__le__subst2,axiom,
! [A2: risk_Free_account,B: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_960_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_961_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_962_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_963_order__less__le__subst2,axiom,
! [A2: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_964_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_965_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_974_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_975_top__greatest,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% top_greatest
thf(fact_976_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_977_top__greatest,axiom,
! [A2: set_nat_real] : ( ord_le2908806416726583473t_real @ A2 @ top_top_set_nat_real ) ).
% top_greatest
thf(fact_978_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_979_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_980_top_Oextremum__unique,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
= ( A2 = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_981_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_982_top_Oextremum__unique,axiom,
! [A2: set_nat_real] :
( ( ord_le2908806416726583473t_real @ top_top_set_nat_real @ A2 )
= ( A2 = top_top_set_nat_real ) ) ).
% top.extremum_unique
thf(fact_983_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_984_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_985_top_Oextremum__uniqueI,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
=> ( A2 = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_986_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_987_top_Oextremum__uniqueI,axiom,
! [A2: set_nat_real] :
( ( ord_le2908806416726583473t_real @ top_top_set_nat_real @ A2 )
=> ( A2 = top_top_set_nat_real ) ) ).
% top.extremum_uniqueI
thf(fact_988_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_989_le__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_minus_iff
thf(fact_990_le__minus__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ ( uminus3377898441596595772ccount @ B ) )
= ( ord_le4245800335709223507ccount @ B @ ( uminus3377898441596595772ccount @ A2 ) ) ) ).
% le_minus_iff
thf(fact_991_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_992_minus__le__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_993_minus__le__iff,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A2 ) @ B )
= ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_994_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_995_le__imp__neg__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_996_le__imp__neg__le,axiom,
! [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
=> ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B ) @ ( uminus3377898441596595772ccount @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_997_finite__has__minimal2,axiom,
! [A: set_nat_real,A2: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A2 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ X2 @ A2 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_998_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_999_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1000_finite__has__minimal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1001_finite__has__minimal2,axiom,
! [A: set_Ri1641125681238393385ccount,A2: risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ( member5612106785598075018ccount @ A2 @ A )
=> ? [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
& ( ord_le4245800335709223507ccount @ X2 @ A2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A )
=> ( ( ord_le4245800335709223507ccount @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1002_finite__has__maximal2,axiom,
! [A: set_nat_real,A2: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A2 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ A2 @ X2 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1003_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A2 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1004_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1005_finite__has__maximal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ A2 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1006_finite__has__maximal2,axiom,
! [A: set_Ri1641125681238393385ccount,A2: risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ( member5612106785598075018ccount @ A2 @ A )
=> ? [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
& ( ord_le4245800335709223507ccount @ A2 @ X2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A )
=> ( ( ord_le4245800335709223507ccount @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1007_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_1008_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1009_ex__le__of__int,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_le_of_int
thf(fact_1010_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_1011_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_1012_sum__mono,axiom,
! [K3: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K3 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K3 ) @ ( groups1932886352136224148al_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1013_sum__mono,axiom,
! [K3: set_int,F: int > int,G: int > int] :
( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K3 ) @ ( groups4538972089207619220nt_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1014_sum__mono,axiom,
! [K3: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K3 ) @ ( groups3539618377306564664at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1015_sum__mono,axiom,
! [K3: set_a,F: a > int,G: a > int] :
( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ K3 ) @ ( groups6332066207828071664_a_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1016_sum__mono,axiom,
! [K3: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K3 ) @ ( groups1935376822645274424al_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1017_sum__mono,axiom,
! [K3: set_int,F: int > nat,G: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K3 ) @ ( groups4541462559716669496nt_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1018_sum__mono,axiom,
! [K3: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1019_sum__mono,axiom,
! [K3: set_a,F: a > nat,G: a > nat] :
( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K3 ) @ ( groups6334556678337121940_a_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1020_sum__mono,axiom,
! [K3: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K3 ) @ ( groups8097168146408367636l_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1021_sum__mono,axiom,
! [K3: set_int,F: int > real,G: int > real] :
( ! [I3: int] :
( ( member_int @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ K3 ) @ ( groups8778361861064173332t_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1022_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_1023_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D: real,E: real] :
( ( ord_less_real @ D @ E )
=> ( ( P @ D )
=> ( 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_1024_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_1025_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1026_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1027_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1028_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_1029_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_1030_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1031_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1032_eq__nat__nat__iff,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z4 ) )
= ( Z = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1033_int__zle__neg,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1034_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_1035_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1036_int__eq__iff,axiom,
! [M3: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M3 )
= Z )
= ( ( M3
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1037_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1038_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1039_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1040_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1041_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A2 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1042_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1043_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1044_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_1045_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_1046_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_1047_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_1048_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1049_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1050_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1051_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1052_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1053_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1054_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1055_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1056_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1057_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1058_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_1059_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1060_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1061_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1062_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1063_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1064_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1065_zle__int,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% zle_int
thf(fact_1066_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_1067_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1068_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1069_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1070_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1071_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1072_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1073_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M3: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1074_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z5: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ 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_1075_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_1076_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_1077_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1078_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_1079_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1080_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1081_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_1082_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1083_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1084_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1085_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_1086_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1087_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1088_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1089_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1090_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1091_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1092_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1093_diff__less__mono2,axiom,
! [M3: nat,N: nat,L3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L3 )
=> ( ord_less_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1094_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1095_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1096_diff__le__mono2,axiom,
! [M3: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1097_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1098_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1099_diff__le__mono,axiom,
! [M3: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L3 ) @ ( minus_minus_nat @ N @ L3 ) ) ) ).
% diff_le_mono
thf(fact_1100_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1101_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1102_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1103_int__diff__cases,axiom,
! [Z: int] :
~ ! [M5: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1104_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1105_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1106_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1107_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1108_int__minus,axiom,
! [N: nat,M3: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M3 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ) ).
% int_minus
thf(fact_1109_int__ops_I6_J,axiom,
! [A2: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1110_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_1111_nat__diff__distrib,axiom,
! [Z4: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1112_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_1113_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_1114_less__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1115_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1116_minus__int__code_I2_J,axiom,
! [L3: int] :
( ( minus_minus_int @ zero_zero_int @ L3 )
= ( uminus_uminus_int @ L3 ) ) ).
% minus_int_code(2)
thf(fact_1117_diff__nat__eq__if,axiom,
! [Z4: int,Z: int] :
( ( ( ord_less_int @ Z4 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z4 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z4 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z4 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1118_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A4: real,B2: real,C3: real] :
( ( P @ A4 @ B2 )
=> ( ( P @ B2 @ C3 )
=> ( ( ord_less_eq_real @ A4 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C3 )
=> ( P @ A4 @ C3 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B2: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_eq_real @ X2 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B2 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1119_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1120_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1121_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1122_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1123_Suc__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1124_Suc__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_eq
thf(fact_1125_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_1126_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1127_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1128_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1129_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1130_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1131_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1132_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1133_negative__zless,axiom,
! [N: nat,M3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zless
thf(fact_1134_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1135_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1136_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1137_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1138_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1139_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1140_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1141_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1142_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1143_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1144_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1145_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_1146_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1147_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1148_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1149_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1150_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1151_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1152_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1153_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_1154_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1155_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1156_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1157_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1158_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1159_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1160_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1161_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1162_Suc__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_lessD
thf(fact_1163_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1164_Suc__lessI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ( suc @ M3 )
!= N )
=> ( ord_less_nat @ ( suc @ M3 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1165_less__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M3 @ N )
=> ( M3 = N ) ) ) ).
% less_SucE
thf(fact_1166_less__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1167_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1168_less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ).
% less_Suc_eq
thf(fact_1169_not__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_nat @ M3 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1170_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1171_Suc__less__eq2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1172_less__antisym,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
=> ( M3 = N ) ) ) ).
% less_antisym
thf(fact_1173_Suc__less__SucD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_SucD
thf(fact_1174_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1175_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K4 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1176_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1177_not__less__less__Suc__eq,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1178_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1179_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1180_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1181_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1182_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( M3 = zero_zero_nat )
| ? [J2: nat] :
( ( M3
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1183_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_1184_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_1185_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1186_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1187_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_1188_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1189_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1190_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1191_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1192_Suc__diff__Suc,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1193_diff__less__Suc,axiom,
! [M3: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1194_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1195_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1196_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1197_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1198_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1199_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1200_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1201_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1202_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1203_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1204_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1205_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1206_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1207_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D: real,E: real] :
( ( ord_less_real @ D @ E )
=> ( ( P @ D )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1208_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1209_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1210_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1211_zabs__def,axiom,
( abs_abs_int
= ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% zabs_def
thf(fact_1212_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N2 ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% less_eq_account_def
thf(fact_1213_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_1214_nat__abs__int__diff,axiom,
! [A2: nat,B: nat] :
( ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A2 ) ) )
& ( ~ ( ord_less_eq_nat @ A2 @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A2 @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1215_nat__intermed__int__val,axiom,
! [M3: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M3 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_int @ ( F @ M3 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M3 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1216_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1217_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1218_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1219_add__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc_right
thf(fact_1220_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1221_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1222_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1223_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1224_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1225_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1226_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1227_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1228_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1229_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1230_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1231_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1232_less__imp__Suc__add,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1233_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1234_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1235_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1236_less__natE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M3 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1237_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1238_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1239_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1240_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1241_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1242_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1243_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1244_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1245_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1246_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1247_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1248_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1249_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1250_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1251_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1252_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1253_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1254_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1255_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1256_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1257_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ) ).
% add_le_mono
thf(fact_1258_le__Suc__ex,axiom,
! [K: nat,L3: nat] :
( ( ord_less_eq_nat @ K @ L3 )
=> ? [N3: nat] :
( L3
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1259_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1260_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_1261_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_1262_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_1263_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
% Helper facts (9)
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_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( risk_Free_balanced_a @ l @ c )
= ( ( ( groups2740460157737275248a_real
@ ^ [A3: a] : ( risk_F1914734008469130493eserve @ ( l @ A3 ) )
@ top_top_set_a )
= c )
& ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups2740460157737275248a_real
@ ^ [A3: a] : ( risk_F170160801229183585ccount @ ( l @ A3 ) @ N2 )
@ top_top_set_a )
= zero_zero_real ) ) ) ) ).
%------------------------------------------------------------------------------