TPTP Problem File: SLH0556^1.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_00345_010755__5782068_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1550 ( 613 unt; 278 typ; 0 def)
% Number of atoms : 4033 (1358 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11021 ( 252 ~; 57 |; 399 &;8539 @)
% ( 0 <=>;1774 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 1231 (1231 >; 0 *; 0 +; 0 <<)
% Number of symbols : 240 ( 237 usr; 43 con; 0-3 aty)
% Number of variables : 3778 ( 553 ^;3118 !; 107 ?;3778 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:56:40.725
%------------------------------------------------------------------------------
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
set_or1122926678442080148t_real: ( nat > real ) > set_nat_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Product____Type__Ounit,type,
set_or8621721900777396692t_unit: product_unit > set_Product_unit ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Risk____Free____Lending__Oaccount,type,
set_or3854930313887350124ccount: risk_Free_account > set_Ri1641125681238393385ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or58775011639299419et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Product____Type__Ounit_J,type,
set_or2827140217781692084t_unit: set_Product_unit > set_set_Product_unit ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or5092868708245317595t_real: set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_or2377417256128143778ccount: set_Ri1641125681238393385ccount > set_se7412911595254129503ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_or4720407823748415505et_int: set_set_int > set_set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7210490968680142261et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
set_or3722988303374331796t_unit: set_set_Product_unit > set_se7118615804146677933t_unit ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4016371710855203973_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
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__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Risk____Free____Lending__Oaccount,type,
member5612106785598075018ccount: risk_Free_account > set_Ri1641125681238393385ccount > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
member8751979616678796480ccount: set_Ri1641125681238393385ccount > set_se7412911595254129503ccount > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $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_assms_I1_J,axiom,
ord_less_eq_real @ zero_zero_real @ c ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
risk_F6417547810208190312nced_a @ l @ c ).
% assms(2)
thf(fact_2_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_3_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [Uu: real] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_4_sum_Oneutral__const,axiom,
! [A: set_nat_real] :
( ( groups383684539861946442ccount
@ ^ [Uu: nat > real] : zero_z1425366712893667068ccount
@ A )
= zero_z1425366712893667068ccount ) ).
% sum.neutral_const
thf(fact_5_sum_Oneutral__const,axiom,
! [A: set_nat_real] :
( ( groups4253619806861319043l_real
@ ^ [Uu: nat > real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_6_sum_Oneutral__const,axiom,
! [A: set_nat_real] :
( ( groups780007972294800423al_nat
@ ^ [Uu: nat > real] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_7_sum_Oneutral__const,axiom,
! [A: set_nat_real] :
( ( groups777517501785750147al_int
@ ^ [Uu: nat > real] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_8_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_9_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [Uu: a] : zero_z1425366712893667068ccount
@ A )
= zero_z1425366712893667068ccount ) ).
% sum.neutral_const
thf(fact_10_atMost__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I @ K ) ) ).
% atMost_iff
thf(fact_11_atMost__iff,axiom,
! [I: risk_Free_account,K: risk_Free_account] :
( ( member5612106785598075018ccount @ I @ ( set_or3854930313887350124ccount @ K ) )
= ( ord_le4245800335709223507ccount @ I @ K ) ) ).
% atMost_iff
thf(fact_12_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_13_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_14_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_15_atMost__iff,axiom,
! [I: nat > real,K: nat > real] :
( ( member_nat_real @ I @ ( set_or1122926678442080148t_real @ K ) )
= ( ord_less_eq_nat_real @ I @ K ) ) ).
% atMost_iff
thf(fact_16_atMost__iff,axiom,
! [I: set_Ri1641125681238393385ccount,K: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ I @ ( set_or2377417256128143778ccount @ K ) )
= ( ord_le4487465848215339657ccount @ I @ K ) ) ).
% atMost_iff
thf(fact_17_atMost__iff,axiom,
! [I: set_real,K: set_real] :
( ( member_set_real @ I @ ( set_or5092868708245317595t_real @ K ) )
= ( ord_less_eq_set_real @ I @ K ) ) ).
% atMost_iff
thf(fact_18_atMost__iff,axiom,
! [I: set_int,K: set_int] :
( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
= ( ord_less_eq_set_int @ I @ K ) ) ).
% atMost_iff
thf(fact_19_atMost__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_20_atMost__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_21_atMost__subset__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or3854930313887350124ccount @ X ) @ ( set_or3854930313887350124ccount @ Y ) )
= ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_22_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_23_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_24_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_25_atMost__subset__iff,axiom,
! [X: set_Ri1641125681238393385ccount,Y: set_Ri1641125681238393385ccount] :
( ( ord_le5934089292365423551ccount @ ( set_or2377417256128143778ccount @ X ) @ ( set_or2377417256128143778ccount @ Y ) )
= ( ord_le4487465848215339657ccount @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_26_atMost__subset__iff,axiom,
! [X: set_real,Y: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or5092868708245317595t_real @ X ) @ ( set_or5092868708245317595t_real @ Y ) )
= ( ord_less_eq_set_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_27_atMost__subset__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_28_atMost__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_29_atMost__subset__iff,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or7210490968680142261et_nat @ X ) @ ( set_or7210490968680142261et_nat @ Y ) )
= ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_30_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_31_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_32_atMost__eq__UNIV__iff,axiom,
! [X: set_set_Product_unit] :
( ( ( set_or3722988303374331796t_unit @ X )
= top_to2430762689449481725t_unit )
= ( X = top_to1767297665138865437t_unit ) ) ).
% atMost_eq_UNIV_iff
thf(fact_33_atMost__eq__UNIV__iff,axiom,
! [X: set_set_int] :
( ( ( set_or4720407823748415505et_int @ X )
= top_to5524576366173240574et_int )
= ( X = top_top_set_set_int ) ) ).
% atMost_eq_UNIV_iff
thf(fact_34_atMost__eq__UNIV__iff,axiom,
! [X: set_set_nat] :
( ( ( set_or7210490968680142261et_nat @ X )
= top_to1114752748106383522et_nat )
= ( X = top_top_set_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_35_atMost__eq__UNIV__iff,axiom,
! [X: set_set_a] :
( ( ( set_or4016371710855203973_set_a @ X )
= top_to4027821306633060462_set_a )
= ( X = top_top_set_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_36_atMost__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= top_top_set_set_a )
= ( X = top_top_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_37_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_38_atMost__eq__UNIV__iff,axiom,
! [X: set_int] :
( ( ( set_or58775011639299419et_int @ X )
= top_top_set_set_int )
= ( X = top_top_set_int ) ) ).
% atMost_eq_UNIV_iff
thf(fact_39_atMost__eq__UNIV__iff,axiom,
! [X: set_Product_unit] :
( ( ( set_or2827140217781692084t_unit @ X )
= top_to1767297665138865437t_unit )
= ( X = top_to1996260823553986621t_unit ) ) ).
% atMost_eq_UNIV_iff
thf(fact_40_atMost__eq__UNIV__iff,axiom,
! [X: product_unit] :
( ( ( set_or8621721900777396692t_unit @ X )
= top_to1996260823553986621t_unit )
= ( X = top_top_Product_unit ) ) ).
% atMost_eq_UNIV_iff
thf(fact_41_sum__nonneg,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_42_sum__nonneg,axiom,
! [A: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups8516999891779824987ccount @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_43_sum__nonneg,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_44_sum__nonneg,axiom,
! [A: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_45_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_46_sum__nonneg,axiom,
! [A: set_a,F: a > risk_Free_account] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups4655409347963886775ccount @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_47_sum__nonneg,axiom,
! [A: set_nat_real,F: ( nat > real ) > real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups4253619806861319043l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_48_sum__nonneg,axiom,
! [A: set_nat_real,F: ( nat > real ) > risk_Free_account] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups383684539861946442ccount @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_49_sum__nonneg,axiom,
! [A: set_nat_real,F: ( nat > real ) > nat] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups780007972294800423al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_50_sum__nonneg,axiom,
! [A: set_nat_real,F: ( nat > real ) > int] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups777517501785750147al_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_51_sum__nonpos,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_52_sum__nonpos,axiom,
! [A: set_int,F: int > real] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_53_sum__nonpos,axiom,
! [A: set_Ri1641125681238393385ccount,F: risk_Free_account > real] :
( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8212752698507433307t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_54_sum__nonpos,axiom,
! [A: set_a,F: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_55_sum__nonpos,axiom,
! [A: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_56_sum__nonpos,axiom,
! [A: set_nat,F: nat > risk_Free_account] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_57_sum__nonpos,axiom,
! [A: set_int,F: int > risk_Free_account] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups2220918773033463387ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_58_sum__nonpos,axiom,
! [A: set_Ri1641125681238393385ccount,F: risk_Free_account > risk_Free_account] :
( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups5726723449141454370ccount @ F @ A ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_59_sum__nonpos,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_60_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_61_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_62_atMost__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= ( set_or4236626031148496127et_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_63_atMost__eq__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( set_or58775011639299419et_int @ X )
= ( set_or58775011639299419et_int @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_64_atMost__eq__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= ( set_ord_atMost_set_a @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_65_atMost__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_atMost_int @ X )
= ( set_ord_atMost_int @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_66_atMost__eq__iff,axiom,
! [X: nat > real,Y: nat > real] :
( ( ( set_or1122926678442080148t_real @ X )
= ( set_or1122926678442080148t_real @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_67_balanced__def,axiom,
( risk_F3851798905806922871t_unit
= ( ^ [L: product_unit > risk_Free_account,C: real] :
( ( groups7985565240515220678ccount @ L @ top_to1996260823553986621t_unit )
= ( risk_Free_just_cash @ C ) ) ) ) ).
% balanced_def
thf(fact_68_balanced__def,axiom,
( risk_F53418252958066007t_unit
= ( ^ [L: set_Product_unit > risk_Free_account,C: real] :
( ( groups6751574267225893286ccount @ L @ top_to1767297665138865437t_unit )
= ( risk_Free_just_cash @ C ) ) ) ) ).
% balanced_def
thf(fact_69_balanced__def,axiom,
( risk_F7528305526176780488_set_a
= ( ^ [L: set_a > risk_Free_account,C: real] :
( ( groups5494456780414153239ccount @ L @ top_top_set_set_a )
= ( risk_Free_just_cash @ C ) ) ) ) ).
% balanced_def
thf(fact_70_balanced__def,axiom,
( risk_F6417547810208190312nced_a
= ( ^ [L: a > risk_Free_account,C: real] :
( ( groups4655409347963886775ccount @ L @ top_top_set_a )
= ( risk_Free_just_cash @ C ) ) ) ) ).
% balanced_def
thf(fact_71_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_72_UNIV__I,axiom,
! [X: risk_Free_account] : ( member5612106785598075018ccount @ X @ top_to4387612366039908569ccount ) ).
% UNIV_I
thf(fact_73_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_74_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_75_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_76_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_77_UNIV__I,axiom,
! [X: set_Product_unit] : ( member5877623283571906838t_unit @ X @ top_to1767297665138865437t_unit ) ).
% UNIV_I
thf(fact_78_UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% UNIV_I
thf(fact_79_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_80_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_81_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_82_iso__tuple__UNIV__I,axiom,
! [X: risk_Free_account] : ( member5612106785598075018ccount @ X @ top_to4387612366039908569ccount ) ).
% iso_tuple_UNIV_I
thf(fact_83_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_84_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_85_iso__tuple__UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_86_iso__tuple__UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_87_iso__tuple__UNIV__I,axiom,
! [X: set_Product_unit] : ( member5877623283571906838t_unit @ X @ top_to1767297665138865437t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_88_iso__tuple__UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_89_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_90_iso__tuple__UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_91_atMost__UNIV__triv,axiom,
( ( set_or2827140217781692084t_unit @ top_to1996260823553986621t_unit )
= top_to1767297665138865437t_unit ) ).
% atMost_UNIV_triv
thf(fact_92_atMost__UNIV__triv,axiom,
( ( set_or3722988303374331796t_unit @ top_to1767297665138865437t_unit )
= top_to2430762689449481725t_unit ) ).
% atMost_UNIV_triv
thf(fact_93_atMost__UNIV__triv,axiom,
( ( set_or4720407823748415505et_int @ top_top_set_set_int )
= top_to5524576366173240574et_int ) ).
% atMost_UNIV_triv
thf(fact_94_atMost__UNIV__triv,axiom,
( ( set_or7210490968680142261et_nat @ top_top_set_set_nat )
= top_to1114752748106383522et_nat ) ).
% atMost_UNIV_triv
thf(fact_95_atMost__UNIV__triv,axiom,
( ( set_or4016371710855203973_set_a @ top_top_set_set_a )
= top_to4027821306633060462_set_a ) ).
% atMost_UNIV_triv
thf(fact_96_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_97_atMost__UNIV__triv,axiom,
( ( set_or58775011639299419et_int @ top_top_set_int )
= top_top_set_set_int ) ).
% atMost_UNIV_triv
thf(fact_98_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_a @ top_top_set_a )
= top_top_set_set_a ) ).
% atMost_UNIV_triv
thf(fact_99_subsetI,axiom,
! [A: set_nat_real,B: set_nat_real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( member_nat_real @ X2 @ B ) )
=> ( ord_le2908806416726583473t_real @ A @ B ) ) ).
% subsetI
thf(fact_100_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_101_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_102_subsetI,axiom,
! [A: set_int,B: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B ) )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% subsetI
thf(fact_103_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_104_subsetI,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount] :
( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( member5612106785598075018ccount @ X2 @ B ) )
=> ( ord_le4487465848215339657ccount @ A @ B ) ) ).
% subsetI
thf(fact_105_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_106_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_107_subset__antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_108_subset__antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_109_subset__antisym,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A @ B )
=> ( ( ord_le4487465848215339657ccount @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_110_subset__antisym,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_111_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_112_Rep__account__just__cash,axiom,
! [C2: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C2 ) )
= ( ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_113_in__mono,axiom,
! [A: set_nat_real,B: set_nat_real,X: nat > real] :
( ( ord_le2908806416726583473t_real @ A @ B )
=> ( ( member_nat_real @ X @ A )
=> ( member_nat_real @ X @ B ) ) ) ).
% in_mono
thf(fact_114_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_115_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_116_in__mono,axiom,
! [A: set_int,B: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( member_int @ X @ A )
=> ( member_int @ X @ B ) ) ) ).
% in_mono
thf(fact_117_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_118_in__mono,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount,X: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ A @ B )
=> ( ( member5612106785598075018ccount @ X @ A )
=> ( member5612106785598075018ccount @ X @ B ) ) ) ).
% in_mono
thf(fact_119_in__mono,axiom,
! [A: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_120_subsetD,axiom,
! [A: set_nat_real,B: set_nat_real,C2: nat > real] :
( ( ord_le2908806416726583473t_real @ A @ B )
=> ( ( member_nat_real @ C2 @ A )
=> ( member_nat_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_121_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_122_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_123_subsetD,axiom,
! [A: set_int,B: set_int,C2: int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B ) ) ) ).
% subsetD
thf(fact_124_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_125_subsetD,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount,C2: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ A @ B )
=> ( ( member5612106785598075018ccount @ C2 @ A )
=> ( member5612106785598075018ccount @ C2 @ B ) ) ) ).
% subsetD
thf(fact_126_subsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_127_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_128_equalityE,axiom,
! [A: set_int,B: set_int] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_int @ A @ B )
=> ~ ( ord_less_eq_set_int @ B @ A ) ) ) ).
% equalityE
thf(fact_129_equalityE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_130_equalityE,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount] :
( ( A = B )
=> ~ ( ( ord_le4487465848215339657ccount @ A @ B )
=> ~ ( ord_le4487465848215339657ccount @ B @ A ) ) ) ).
% equalityE
thf(fact_131_equalityE,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_real @ A @ B )
=> ~ ( ord_less_eq_set_real @ B @ A ) ) ) ).
% equalityE
thf(fact_132_subset__eq,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A2: set_nat_real,B2: set_nat_real] :
! [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
=> ( member_nat_real @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_133_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_134_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_135_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A2: set_int,B2: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_136_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B2: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_137_subset__eq,axiom,
( ord_le4487465848215339657ccount
= ( ^ [A2: set_Ri1641125681238393385ccount,B2: set_Ri1641125681238393385ccount] :
! [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
=> ( member5612106785598075018ccount @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_138_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B2: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_139_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_140_equalityD1,axiom,
! [A: set_int,B: set_int] :
( ( A = B )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% equalityD1
thf(fact_141_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_142_equalityD1,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount] :
( ( A = B )
=> ( ord_le4487465848215339657ccount @ A @ B ) ) ).
% equalityD1
thf(fact_143_equalityD1,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% equalityD1
thf(fact_144_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_145_equalityD2,axiom,
! [A: set_int,B: set_int] :
( ( A = B )
=> ( ord_less_eq_set_int @ B @ A ) ) ).
% equalityD2
thf(fact_146_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_147_equalityD2,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount] :
( ( A = B )
=> ( ord_le4487465848215339657ccount @ B @ A ) ) ).
% equalityD2
thf(fact_148_equalityD2,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ B @ A ) ) ).
% equalityD2
thf(fact_149_subset__iff,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A2: set_nat_real,B2: set_nat_real] :
! [T: nat > real] :
( ( member_nat_real @ T @ A2 )
=> ( member_nat_real @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_150_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A2 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_151_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A2 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_152_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A2: set_int,B2: set_int] :
! [T: int] :
( ( member_int @ T @ A2 )
=> ( member_int @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_153_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B2: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A2 )
=> ( member_set_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_154_subset__iff,axiom,
( ord_le4487465848215339657ccount
= ( ^ [A2: set_Ri1641125681238393385ccount,B2: set_Ri1641125681238393385ccount] :
! [T: risk_Free_account] :
( ( member5612106785598075018ccount @ T @ A2 )
=> ( member5612106785598075018ccount @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_155_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B2: set_real] :
! [T: real] :
( ( member_real @ T @ A2 )
=> ( member_real @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_156_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_157_subset__refl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% subset_refl
thf(fact_158_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_159_subset__refl,axiom,
! [A: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ A @ A ) ).
% subset_refl
thf(fact_160_subset__refl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% subset_refl
thf(fact_161_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_162_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_163_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_164_Collect__mono,axiom,
! [P: risk_Free_account > $o,Q: risk_Free_account > $o] :
( ! [X2: risk_Free_account] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le4487465848215339657ccount @ ( collec1856553087948576712ccount @ P ) @ ( collec1856553087948576712ccount @ Q ) ) ) ).
% Collect_mono
thf(fact_165_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_166_subset__trans,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_167_subset__trans,axiom,
! [A: set_int,B: set_int,C3: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C3 )
=> ( ord_less_eq_set_int @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_168_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C3 )
=> ( ord_le6893508408891458716et_nat @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_169_subset__trans,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_Ri1641125681238393385ccount,C3: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A @ B )
=> ( ( ord_le4487465848215339657ccount @ B @ C3 )
=> ( ord_le4487465848215339657ccount @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_170_subset__trans,axiom,
! [A: set_real,B: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ C3 )
=> ( ord_less_eq_set_real @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_171_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_172_set__eq__subset,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
& ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_173_set__eq__subset,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_174_set__eq__subset,axiom,
( ( ^ [Y2: set_Ri1641125681238393385ccount,Z: set_Ri1641125681238393385ccount] : ( Y2 = Z ) )
= ( ^ [A2: set_Ri1641125681238393385ccount,B2: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A2 @ B2 )
& ( ord_le4487465848215339657ccount @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_175_set__eq__subset,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
& ( ord_less_eq_set_real @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_176_Collect__subset,axiom,
! [A: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_177_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_178_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_179_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_180_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_181_Collect__subset,axiom,
! [A: set_Ri1641125681238393385ccount,P: risk_Free_account > $o] :
( ord_le4487465848215339657ccount
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_182_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_183_less__eq__set__def,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A2: set_nat_real,B2: set_nat_real] :
( ord_le7676461544873280788real_o
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ A2 )
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_184_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_185_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_186_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A2: set_int,B2: set_int] :
( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A2 )
@ ^ [X3: int] : ( member_int @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_187_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B2: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_188_less__eq__set__def,axiom,
( ord_le4487465848215339657ccount
= ( ^ [A2: set_Ri1641125681238393385ccount,B2: set_Ri1641125681238393385ccount] :
( ord_le1296111392916142396ount_o
@ ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ A2 )
@ ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_189_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B2: set_real] :
( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A2 )
@ ^ [X3: real] : ( member_real @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_190_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_191_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_192_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_193_Collect__mono__iff,axiom,
! [P: risk_Free_account > $o,Q: risk_Free_account > $o] :
( ( ord_le4487465848215339657ccount @ ( collec1856553087948576712ccount @ P ) @ ( collec1856553087948576712ccount @ Q ) )
= ( ! [X3: risk_Free_account] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_194_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X3: real] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_195_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_196_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_197_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_198_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_199_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_200_top__set__def,axiom,
( top_to1767297665138865437t_unit
= ( collec7787489603430924120t_unit @ top_to5616354022731438592unit_o ) ) ).
% top_set_def
thf(fact_201_top__set__def,axiom,
( top_top_set_set_int
= ( collect_set_int @ top_top_set_int_o ) ) ).
% top_set_def
thf(fact_202_top__set__def,axiom,
( top_top_set_set_nat
= ( collect_set_nat @ top_top_set_nat_o ) ) ).
% top_set_def
thf(fact_203_top__set__def,axiom,
( top_top_set_set_a
= ( collect_set_a @ top_top_set_a_o ) ) ).
% top_set_def
thf(fact_204_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_205_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_206_subset__UNIV,axiom,
! [A: set_set_Product_unit] : ( ord_le3674001228145610605t_unit @ A @ top_to1767297665138865437t_unit ) ).
% subset_UNIV
thf(fact_207_subset__UNIV,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ A @ top_top_set_set_int ) ).
% subset_UNIV
thf(fact_208_subset__UNIV,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_209_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_210_subset__UNIV,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% subset_UNIV
thf(fact_211_subset__UNIV,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ top_top_set_set_nat ) ).
% subset_UNIV
thf(fact_212_subset__UNIV,axiom,
! [A: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ A @ top_to4387612366039908569ccount ) ).
% subset_UNIV
thf(fact_213_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_214_not__UNIV__le__Iic,axiom,
! [H: real] :
~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_ord_atMost_real @ H ) ) ).
% not_UNIV_le_Iic
thf(fact_215_not__UNIV__le__Iic,axiom,
! [H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_le_Iic
thf(fact_216_not__UNIV__le__Iic,axiom,
! [H: int] :
~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_ord_atMost_int @ H ) ) ).
% not_UNIV_le_Iic
thf(fact_217_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_218_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_219_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_220_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_221_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_222_UNIV__witness,axiom,
? [X2: risk_Free_account] : ( member5612106785598075018ccount @ X2 @ top_to4387612366039908569ccount ) ).
% UNIV_witness
thf(fact_223_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_224_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_225_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_226_UNIV__witness,axiom,
? [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_227_UNIV__witness,axiom,
? [X2: set_Product_unit] : ( member5877623283571906838t_unit @ X2 @ top_to1767297665138865437t_unit ) ).
% UNIV_witness
thf(fact_228_UNIV__witness,axiom,
? [X2: set_int] : ( member_set_int @ X2 @ top_top_set_set_int ) ).
% UNIV_witness
thf(fact_229_UNIV__witness,axiom,
? [X2: set_nat] : ( member_set_nat @ X2 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_230_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_231_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_232_UNIV__eq__I,axiom,
! [A: set_Ri1641125681238393385ccount] :
( ! [X2: risk_Free_account] : ( member5612106785598075018ccount @ X2 @ A )
=> ( top_to4387612366039908569ccount = A ) ) ).
% UNIV_eq_I
thf(fact_233_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_234_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_235_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_236_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X2: product_unit] : ( member_Product_unit @ X2 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_237_UNIV__eq__I,axiom,
! [A: set_set_Product_unit] :
( ! [X2: set_Product_unit] : ( member5877623283571906838t_unit @ X2 @ A )
=> ( top_to1767297665138865437t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_238_UNIV__eq__I,axiom,
! [A: set_set_int] :
( ! [X2: set_int] : ( member_set_int @ X2 @ A )
=> ( top_top_set_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_239_UNIV__eq__I,axiom,
! [A: set_set_nat] :
( ! [X2: set_nat] : ( member_set_nat @ X2 @ A )
=> ( top_top_set_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_240_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_241_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T2: set_nat,H: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_242_sum_Oreindex__bij__witness,axiom,
! [S: set_int,I: nat > int,J: int > nat,T2: set_nat,H: nat > real,G: int > real] :
( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( member_int @ ( I @ B3 ) @ S ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_243_sum_Oreindex__bij__witness,axiom,
! [S: set_Ri1641125681238393385ccount,I: nat > risk_Free_account,J: risk_Free_account > nat,T2: set_nat,H: nat > real,G: risk_Free_account > real] :
( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( member5612106785598075018ccount @ ( I @ B3 ) @ S ) )
=> ( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8212752698507433307t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_244_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I: nat > a,J: a > nat,T2: set_nat,H: nat > real,G: a > real] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( member_a @ ( I @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_245_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: a > real,J: real > a,T2: set_a,H: a > risk_Free_account,G: real > risk_Free_account] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_246_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: a > nat,J: nat > a,T2: set_a,H: a > risk_Free_account,G: nat > risk_Free_account] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_247_sum_Oreindex__bij__witness,axiom,
! [S: set_int,I: a > int,J: int > a,T2: set_a,H: a > risk_Free_account,G: int > risk_Free_account] :
( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member_int @ ( I @ B3 ) @ S ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2220918773033463387ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_248_sum_Oreindex__bij__witness,axiom,
! [S: set_Ri1641125681238393385ccount,I: a > risk_Free_account,J: risk_Free_account > a,T2: set_a,H: a > risk_Free_account,G: risk_Free_account > risk_Free_account] :
( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member5612106785598075018ccount @ ( I @ B3 ) @ S ) )
=> ( ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5726723449141454370ccount @ G @ S )
= ( groups4655409347963886775ccount @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_249_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T2: set_real,H: real > int,G: nat > int] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_real @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups1932886352136224148al_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_250_sum_Oreindex__bij__witness,axiom,
! [S: set_int,I: real > int,J: int > real,T2: set_real,H: real > int,G: int > int] :
( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( member_real @ ( J @ A3 ) @ T2 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( member_int @ ( I @ B3 ) @ S ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S )
= ( groups1932886352136224148al_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_251_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( ( member_real @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_252_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > int,A: set_int,H: int > nat,Gamma: nat > real,Phi: int > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( ( member_int @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_253_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > risk_Free_account,A: set_Ri1641125681238393385ccount,H: risk_Free_account > nat,Gamma: nat > real,Phi: risk_Free_account > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( ( member5612106785598075018ccount @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8212752698507433307t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_254_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > a,A: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_255_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > real,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( ( member_real @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_256_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > nat,A: set_nat,H: nat > a,Gamma: a > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_257_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > int,A: set_int,H: int > a,Gamma: a > risk_Free_account,Phi: int > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( ( member_int @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2220918773033463387ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_258_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > risk_Free_account,A: set_Ri1641125681238393385ccount,H: risk_Free_account > a,Gamma: a > risk_Free_account,Phi: risk_Free_account > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( ( member5612106785598075018ccount @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5726723449141454370ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_259_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > int,Phi: nat > int] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_260_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > int,A: set_int,H: int > real,Gamma: real > int,Phi: int > int] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B )
=> ( ( member_int @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_261_sum_Oeq__general,axiom,
! [B: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_262_sum_Oeq__general,axiom,
! [B: set_nat,A: set_int,H: int > nat,Gamma: nat > real,Phi: int > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8778361861064173332t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_263_sum_Oeq__general,axiom,
! [B: set_nat,A: set_Ri1641125681238393385ccount,H: risk_Free_account > nat,Gamma: nat > real,Phi: risk_Free_account > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ? [X4: risk_Free_account] :
( ( member5612106785598075018ccount @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: risk_Free_account] :
( ( ( member5612106785598075018ccount @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8212752698507433307t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_264_sum_Oeq__general,axiom,
! [B: set_nat,A: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_265_sum_Oeq__general,axiom,
! [B: set_a,A: set_real,H: real > a,Gamma: a > risk_Free_account,Phi: real > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8516999891779824987ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_266_sum_Oeq__general,axiom,
! [B: set_a,A: set_nat,H: nat > a,Gamma: a > risk_Free_account,Phi: nat > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6033208628184776703ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_267_sum_Oeq__general,axiom,
! [B: set_a,A: set_int,H: int > a,Gamma: a > risk_Free_account,Phi: int > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2220918773033463387ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_268_sum_Oeq__general,axiom,
! [B: set_a,A: set_Ri1641125681238393385ccount,H: risk_Free_account > a,Gamma: a > risk_Free_account,Phi: risk_Free_account > risk_Free_account] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ? [X4: risk_Free_account] :
( ( member5612106785598075018ccount @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: risk_Free_account] :
( ( ( member5612106785598075018ccount @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ( member_a @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5726723449141454370ccount @ Phi @ A )
= ( groups4655409347963886775ccount @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_269_sum_Oeq__general,axiom,
! [B: set_real,A: set_nat,H: nat > real,Gamma: real > int,Phi: nat > int] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_270_sum_Oeq__general,axiom,
! [B: set_real,A: set_int,H: int > real,Gamma: real > int,Phi: int > int] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: int] :
( ( ( member_int @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4538972089207619220nt_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_271_mem__Collect__eq,axiom,
! [A4: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A4 @ ( collect_nat_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_272_mem__Collect__eq,axiom,
! [A4: real,P: real > $o] :
( ( member_real @ A4 @ ( collect_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_273_mem__Collect__eq,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A4 @ ( collect_set_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_274_mem__Collect__eq,axiom,
! [A4: risk_Free_account,P: risk_Free_account > $o] :
( ( member5612106785598075018ccount @ A4 @ ( collec1856553087948576712ccount @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_275_mem__Collect__eq,axiom,
! [A4: a,P: a > $o] :
( ( member_a @ A4 @ ( collect_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_276_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_277_mem__Collect__eq,axiom,
! [A4: int,P: int > $o] :
( ( member_int @ A4 @ ( collect_int @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_278_Collect__mem__eq,axiom,
! [A: set_nat_real] :
( ( collect_nat_real
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_279_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_280_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_281_Collect__mem__eq,axiom,
! [A: set_Ri1641125681238393385ccount] :
( ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_282_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_283_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_284_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_285_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_286_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_287_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_288_sum_Ocong,axiom,
! [A: set_a,B: set_a,G: a > risk_Free_account,H: a > risk_Free_account] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4655409347963886775ccount @ G @ A )
= ( groups4655409347963886775ccount @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_289_sum_Ocong,axiom,
! [A: set_real,B: set_real,G: real > int,H: real > int] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ A )
= ( groups1932886352136224148al_int @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_290_sum_Ocong,axiom,
! [A: set_nat_real,B: set_nat_real,G: ( nat > real ) > risk_Free_account,H: ( nat > real ) > risk_Free_account] :
( ( A = B )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups383684539861946442ccount @ G @ A )
= ( groups383684539861946442ccount @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_291_sum_Ocong,axiom,
! [A: set_nat_real,B: set_nat_real,G: ( nat > real ) > real,H: ( nat > real ) > real] :
( ( A = B )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4253619806861319043l_real @ G @ A )
= ( groups4253619806861319043l_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_292_sum_Ocong,axiom,
! [A: set_nat_real,B: set_nat_real,G: ( nat > real ) > nat,H: ( nat > real ) > nat] :
( ( A = B )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups780007972294800423al_nat @ G @ A )
= ( groups780007972294800423al_nat @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_293_sum_Ocong,axiom,
! [A: set_nat_real,B: set_nat_real,G: ( nat > real ) > int,H: ( nat > real ) > int] :
( ( A = B )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups777517501785750147al_int @ G @ A )
= ( groups777517501785750147al_int @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_294_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X3: a] : $true ) ) ).
% UNIV_def
thf(fact_295_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X3: nat] : $true ) ) ).
% UNIV_def
thf(fact_296_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X3: int] : $true ) ) ).
% UNIV_def
thf(fact_297_UNIV__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit
@ ^ [X3: product_unit] : $true ) ) ).
% UNIV_def
thf(fact_298_UNIV__def,axiom,
( top_to1767297665138865437t_unit
= ( collec7787489603430924120t_unit
@ ^ [X3: set_Product_unit] : $true ) ) ).
% UNIV_def
thf(fact_299_UNIV__def,axiom,
( top_top_set_set_int
= ( collect_set_int
@ ^ [X3: set_int] : $true ) ) ).
% UNIV_def
thf(fact_300_UNIV__def,axiom,
( top_top_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $true ) ) ).
% UNIV_def
thf(fact_301_UNIV__def,axiom,
( top_top_set_set_a
= ( collect_set_a
@ ^ [X3: set_a] : $true ) ) ).
% UNIV_def
thf(fact_302_sum_Oswap,axiom,
! [G: nat > nat > real,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_303_sum_Oswap,axiom,
! [G: a > a > risk_Free_account,B: set_a,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [I2: a] : ( groups4655409347963886775ccount @ ( G @ I2 ) @ B )
@ A )
= ( groups4655409347963886775ccount
@ ^ [J2: a] :
( groups4655409347963886775ccount
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_304_sum_Oswap,axiom,
! [G: real > real > int,B: set_real,A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [I2: real] : ( groups1932886352136224148al_int @ ( G @ I2 ) @ B )
@ A )
= ( groups1932886352136224148al_int
@ ^ [J2: real] :
( groups1932886352136224148al_int
@ ^ [I2: real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_305_sum_Oswap,axiom,
! [G: nat > ( nat > real ) > real,B: set_nat_real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups4253619806861319043l_real @ ( G @ I2 ) @ B )
@ A )
= ( groups4253619806861319043l_real
@ ^ [J2: nat > real] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_306_sum_Oswap,axiom,
! [G: a > ( nat > real ) > risk_Free_account,B: set_nat_real,A: set_a] :
( ( groups4655409347963886775ccount
@ ^ [I2: a] : ( groups383684539861946442ccount @ ( G @ I2 ) @ B )
@ A )
= ( groups383684539861946442ccount
@ ^ [J2: nat > real] :
( groups4655409347963886775ccount
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_307_sum_Oswap,axiom,
! [G: real > ( nat > real ) > int,B: set_nat_real,A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [I2: real] : ( groups777517501785750147al_int @ ( G @ I2 ) @ B )
@ A )
= ( groups777517501785750147al_int
@ ^ [J2: nat > real] :
( groups1932886352136224148al_int
@ ^ [I2: real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_308_sum_Oswap,axiom,
! [G: ( nat > real ) > a > risk_Free_account,B: set_a,A: set_nat_real] :
( ( groups383684539861946442ccount
@ ^ [I2: nat > real] : ( groups4655409347963886775ccount @ ( G @ I2 ) @ B )
@ A )
= ( groups4655409347963886775ccount
@ ^ [J2: a] :
( groups383684539861946442ccount
@ ^ [I2: nat > real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_309_sum_Oswap,axiom,
! [G: ( nat > real ) > nat > real,B: set_nat,A: set_nat_real] :
( ( groups4253619806861319043l_real
@ ^ [I2: nat > real] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups4253619806861319043l_real
@ ^ [I2: nat > real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_310_sum_Oswap,axiom,
! [G: ( nat > real ) > real > int,B: set_real,A: set_nat_real] :
( ( groups777517501785750147al_int
@ ^ [I2: nat > real] : ( groups1932886352136224148al_int @ ( G @ I2 ) @ B )
@ A )
= ( groups1932886352136224148al_int
@ ^ [J2: real] :
( groups777517501785750147al_int
@ ^ [I2: nat > real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_311_sum_Oswap,axiom,
! [G: ( nat > real ) > ( nat > real ) > risk_Free_account,B: set_nat_real,A: set_nat_real] :
( ( groups383684539861946442ccount
@ ^ [I2: nat > real] : ( groups383684539861946442ccount @ ( G @ I2 ) @ B )
@ A )
= ( groups383684539861946442ccount
@ ^ [J2: nat > real] :
( groups383684539861946442ccount
@ ^ [I2: nat > real] : ( G @ I2 @ J2 )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_312_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_313_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_314_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_315_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_316_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: risk_Free_account > real,A: set_Ri1641125681238393385ccount] :
( ( ( groups8212752698507433307t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_317_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > real,A: set_a] :
( ( ( groups2740460157737275248a_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_318_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_319_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > risk_Free_account,A: set_nat] :
( ( ( groups6033208628184776703ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_320_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > risk_Free_account,A: set_int] :
( ( ( groups2220918773033463387ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_321_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: risk_Free_account > risk_Free_account,A: set_Ri1641125681238393385ccount] :
( ( ( groups5726723449141454370ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: risk_Free_account] :
( ( member5612106785598075018ccount @ A3 @ A )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_322_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_323_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_324_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_325_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_326_sum_Oneutral,axiom,
! [A: set_real,G: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups1932886352136224148al_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_327_sum_Oneutral,axiom,
! [A: set_nat_real,G: ( nat > real ) > risk_Free_account] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups383684539861946442ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.neutral
thf(fact_328_sum_Oneutral,axiom,
! [A: set_nat_real,G: ( nat > real ) > real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups4253619806861319043l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_329_sum_Oneutral,axiom,
! [A: set_nat_real,G: ( nat > real ) > nat] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups780007972294800423al_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_330_sum_Oneutral,axiom,
! [A: set_nat_real,G: ( nat > real ) > int] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups777517501785750147al_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_331_not__UNIV__eq__Iic,axiom,
! [H2: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_eq_Iic
thf(fact_332_not__UNIV__eq__Iic,axiom,
! [H2: int] :
( top_top_set_int
!= ( set_ord_atMost_int @ H2 ) ) ).
% not_UNIV_eq_Iic
thf(fact_333_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_334_sum__mono,axiom,
! [K2: set_int,F: int > real,G: int > real] :
( ! [I3: int] :
( ( member_int @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ K2 ) @ ( groups8778361861064173332t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_335_sum__mono,axiom,
! [K2: set_Ri1641125681238393385ccount,F: risk_Free_account > real,G: risk_Free_account > real] :
( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8212752698507433307t_real @ F @ K2 ) @ ( groups8212752698507433307t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_336_sum__mono,axiom,
! [K2: set_a,F: a > real,G: a > real] :
( ! [I3: a] :
( ( member_a @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ K2 ) @ ( groups2740460157737275248a_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_337_sum__mono,axiom,
! [K2: set_real,F: real > risk_Free_account,G: real > risk_Free_account] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ K2 ) @ ( groups8516999891779824987ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_338_sum__mono,axiom,
! [K2: set_nat,F: nat > risk_Free_account,G: nat > risk_Free_account] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ K2 ) @ ( groups6033208628184776703ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_339_sum__mono,axiom,
! [K2: set_int,F: int > risk_Free_account,G: int > risk_Free_account] :
( ! [I3: int] :
( ( member_int @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups2220918773033463387ccount @ F @ K2 ) @ ( groups2220918773033463387ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_340_sum__mono,axiom,
! [K2: set_Ri1641125681238393385ccount,F: risk_Free_account > risk_Free_account,G: risk_Free_account > risk_Free_account] :
( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups5726723449141454370ccount @ F @ K2 ) @ ( groups5726723449141454370ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_341_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_342_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_343_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_344_atMost__def,axiom,
( set_or3854930313887350124ccount
= ( ^ [U: risk_Free_account] :
( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] : ( ord_le4245800335709223507ccount @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_345_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_346_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_347_atMost__def,axiom,
( set_or2377417256128143778ccount
= ( ^ [U: set_Ri1641125681238393385ccount] :
( collec2171358002926308350ccount
@ ^ [X3: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_348_atMost__def,axiom,
( set_or5092868708245317595t_real
= ( ^ [U: set_real] :
( collect_set_real
@ ^ [X3: set_real] : ( ord_less_eq_set_real @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_349_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_350_atMost__def,axiom,
( set_or58775011639299419et_int
= ( ^ [U: set_int] :
( collect_set_int
@ ^ [X3: set_int] : ( ord_less_eq_set_int @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_351_atMost__def,axiom,
( set_ord_atMost_set_a
= ( ^ [U: set_a] :
( collect_set_a
@ ^ [X3: set_a] : ( ord_less_eq_set_a @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_352_atMost__def,axiom,
( set_or7210490968680142261et_nat
= ( ^ [U: set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_353_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_354_dual__order_Orefl,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_355_dual__order_Orefl,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_356_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_357_dual__order_Orefl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_358_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_359_dual__order_Orefl,axiom,
! [A4: nat > real] : ( ord_less_eq_nat_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_360_dual__order_Orefl,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_361_dual__order_Orefl,axiom,
! [A4: set_set_nat] : ( ord_le6893508408891458716et_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_362_dual__order_Orefl,axiom,
! [A4: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_363_dual__order_Orefl,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_364_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_365_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_366_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_367_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_368_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_369_order__refl,axiom,
! [X: nat > real] : ( ord_less_eq_nat_real @ X @ X ) ).
% order_refl
thf(fact_370_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_371_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_372_order__refl,axiom,
! [X: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ X @ X ) ).
% order_refl
thf(fact_373_order__refl,axiom,
! [X: set_real] : ( ord_less_eq_set_real @ X @ X ) ).
% order_refl
thf(fact_374_top_Oextremum__uniqueI,axiom,
! [A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A4 )
=> ( A4 = top_top_Product_unit ) ) ).
% top.extremum_uniqueI
thf(fact_375_top_Oextremum__uniqueI,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A4 )
=> ( A4 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_376_top_Oextremum__uniqueI,axiom,
! [A4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A4 )
=> ( A4 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_377_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_378_top_Oextremum__uniqueI,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A4 )
=> ( A4 = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_379_top_Oextremum__uniqueI,axiom,
! [A4: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ top_to4387612366039908569ccount @ A4 )
=> ( A4 = top_to4387612366039908569ccount ) ) ).
% top.extremum_uniqueI
thf(fact_380_top_Oextremum__uniqueI,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
=> ( A4 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_381_top_Oextremum__uniqueI,axiom,
! [A4: set_set_Product_unit] :
( ( ord_le3674001228145610605t_unit @ top_to1767297665138865437t_unit @ A4 )
=> ( A4 = top_to1767297665138865437t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_382_top_Oextremum__uniqueI,axiom,
! [A4: set_set_int] :
( ( ord_le4403425263959731960et_int @ top_top_set_set_int @ A4 )
=> ( A4 = top_top_set_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_383_top_Oextremum__uniqueI,axiom,
! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A4 )
=> ( A4 = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_384_top_Oextremum__unique,axiom,
! [A4: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A4 )
= ( A4 = top_top_Product_unit ) ) ).
% top.extremum_unique
thf(fact_385_top_Oextremum__unique,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A4 )
= ( A4 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_386_top_Oextremum__unique,axiom,
! [A4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A4 )
= ( A4 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_387_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_388_top_Oextremum__unique,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A4 )
= ( A4 = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_389_top_Oextremum__unique,axiom,
! [A4: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ top_to4387612366039908569ccount @ A4 )
= ( A4 = top_to4387612366039908569ccount ) ) ).
% top.extremum_unique
thf(fact_390_top_Oextremum__unique,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A4 )
= ( A4 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_391_top_Oextremum__unique,axiom,
! [A4: set_set_Product_unit] :
( ( ord_le3674001228145610605t_unit @ top_to1767297665138865437t_unit @ A4 )
= ( A4 = top_to1767297665138865437t_unit ) ) ).
% top.extremum_unique
thf(fact_392_top_Oextremum__unique,axiom,
! [A4: set_set_int] :
( ( ord_le4403425263959731960et_int @ top_top_set_set_int @ A4 )
= ( A4 = top_top_set_set_int ) ) ).
% top.extremum_unique
thf(fact_393_top_Oextremum__unique,axiom,
! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A4 )
= ( A4 = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_394_top__greatest,axiom,
! [A4: product_unit] : ( ord_le3221252021190050221t_unit @ A4 @ top_top_Product_unit ) ).
% top_greatest
thf(fact_395_top__greatest,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ top_top_set_a ) ).
% top_greatest
thf(fact_396_top__greatest,axiom,
! [A4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A4 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_397_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_398_top__greatest,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ top_top_set_int ) ).
% top_greatest
thf(fact_399_top__greatest,axiom,
! [A4: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ A4 @ top_to4387612366039908569ccount ) ).
% top_greatest
thf(fact_400_top__greatest,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ top_top_set_real ) ).
% top_greatest
thf(fact_401_top__greatest,axiom,
! [A4: set_set_Product_unit] : ( ord_le3674001228145610605t_unit @ A4 @ top_to1767297665138865437t_unit ) ).
% top_greatest
thf(fact_402_top__greatest,axiom,
! [A4: set_set_int] : ( ord_le4403425263959731960et_int @ A4 @ top_top_set_set_int ) ).
% top_greatest
thf(fact_403_top__greatest,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ A4 @ top_top_set_set_a ) ).
% top_greatest
thf(fact_404_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_405_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_406_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_407_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_408_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C: real] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_409_pred__subset__eq,axiom,
! [R: set_nat_real,S: set_nat_real] :
( ( ord_le7676461544873280788real_o
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ R )
@ ^ [X3: nat > real] : ( member_nat_real @ X3 @ S ) )
= ( ord_le2908806416726583473t_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_410_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_411_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_412_pred__subset__eq,axiom,
! [R: set_int,S: set_int] :
( ( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ R )
@ ^ [X3: int] : ( member_int @ X3 @ S ) )
= ( ord_less_eq_set_int @ R @ S ) ) ).
% pred_subset_eq
thf(fact_413_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_414_pred__subset__eq,axiom,
! [R: set_Ri1641125681238393385ccount,S: set_Ri1641125681238393385ccount] :
( ( ord_le1296111392916142396ount_o
@ ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ R )
@ ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ S ) )
= ( ord_le4487465848215339657ccount @ R @ S ) ) ).
% pred_subset_eq
thf(fact_415_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X3: real] : ( member_real @ X3 @ R )
@ ^ [X3: real] : ( member_real @ X3 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_416_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_417_conj__subset__def,axiom,
! [A: set_int,P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ A
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_int @ A @ ( collect_int @ P ) )
& ( ord_less_eq_set_int @ A @ ( collect_int @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_418_conj__subset__def,axiom,
! [A: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P ) )
& ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_419_conj__subset__def,axiom,
! [A: set_Ri1641125681238393385ccount,P: risk_Free_account > $o,Q: risk_Free_account > $o] :
( ( ord_le4487465848215339657ccount @ A
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le4487465848215339657ccount @ A @ ( collec1856553087948576712ccount @ P ) )
& ( ord_le4487465848215339657ccount @ A @ ( collec1856553087948576712ccount @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_420_conj__subset__def,axiom,
! [A: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A
@ ( collect_real
@ ^ [X3: real] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_real @ A @ ( collect_real @ P ) )
& ( ord_less_eq_set_real @ A @ ( collect_real @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_421_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_422_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_423_nle__le,axiom,
! [A4: real,B4: real] :
( ( ~ ( ord_less_eq_real @ A4 @ B4 ) )
= ( ( ord_less_eq_real @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_424_nle__le,axiom,
! [A4: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_425_nle__le,axiom,
! [A4: int,B4: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B4 ) )
= ( ( ord_less_eq_int @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_426_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_427_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_428_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_429_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_430_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [X3: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_431_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_432_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_433_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_434_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat > real,Z: nat > real] : ( Y2 = Z ) )
= ( ^ [X3: nat > real,Y4: nat > real] :
( ( ord_less_eq_nat_real @ X3 @ Y4 )
& ( ord_less_eq_nat_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_435_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [X3: set_int,Y4: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y4 )
& ( ord_less_eq_set_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_436_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
& ( ord_le6893508408891458716et_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_437_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Ri1641125681238393385ccount,Z: set_Ri1641125681238393385ccount] : ( Y2 = Z ) )
= ( ^ [X3: set_Ri1641125681238393385ccount,Y4: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ X3 @ Y4 )
& ( ord_le4487465848215339657ccount @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_438_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [X3: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X3 @ Y4 )
& ( ord_less_eq_set_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_439_ord__eq__le__trans,axiom,
! [A4: real,B4: real,C2: real] :
( ( A4 = B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_440_ord__eq__le__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C2: risk_Free_account] :
( ( A4 = B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_441_ord__eq__le__trans,axiom,
! [A4: nat,B4: nat,C2: nat] :
( ( A4 = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_442_ord__eq__le__trans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( A4 = B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_443_ord__eq__le__trans,axiom,
! [A4: int,B4: int,C2: int] :
( ( A4 = B4 )
=> ( ( ord_less_eq_int @ B4 @ C2 )
=> ( ord_less_eq_int @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_444_ord__eq__le__trans,axiom,
! [A4: nat > real,B4: nat > real,C2: nat > real] :
( ( A4 = B4 )
=> ( ( ord_less_eq_nat_real @ B4 @ C2 )
=> ( ord_less_eq_nat_real @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_445_ord__eq__le__trans,axiom,
! [A4: set_int,B4: set_int,C2: set_int] :
( ( A4 = B4 )
=> ( ( ord_less_eq_set_int @ B4 @ C2 )
=> ( ord_less_eq_set_int @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_446_ord__eq__le__trans,axiom,
! [A4: set_set_nat,B4: set_set_nat,C2: set_set_nat] :
( ( A4 = B4 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_447_ord__eq__le__trans,axiom,
! [A4: set_Ri1641125681238393385ccount,B4: set_Ri1641125681238393385ccount,C2: set_Ri1641125681238393385ccount] :
( ( A4 = B4 )
=> ( ( ord_le4487465848215339657ccount @ B4 @ C2 )
=> ( ord_le4487465848215339657ccount @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_448_ord__eq__le__trans,axiom,
! [A4: set_real,B4: set_real,C2: set_real] :
( ( A4 = B4 )
=> ( ( ord_less_eq_set_real @ B4 @ C2 )
=> ( ord_less_eq_set_real @ A4 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_449_ord__le__eq__trans,axiom,
! [A4: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_450_ord__le__eq__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_451_ord__le__eq__trans,axiom,
! [A4: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_452_ord__le__eq__trans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_453_ord__le__eq__trans,axiom,
! [A4: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_int @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_454_ord__le__eq__trans,axiom,
! [A4: nat > real,B4: nat > real,C2: nat > real] :
( ( ord_less_eq_nat_real @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat_real @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_455_ord__le__eq__trans,axiom,
! [A4: set_int,B4: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_int @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_456_ord__le__eq__trans,axiom,
! [A4: set_set_nat,B4: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_le6893508408891458716et_nat @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_457_ord__le__eq__trans,axiom,
! [A4: set_Ri1641125681238393385ccount,B4: set_Ri1641125681238393385ccount,C2: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_le4487465848215339657ccount @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_458_ord__le__eq__trans,axiom,
! [A4: set_real,B4: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_real @ A4 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_459_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_460_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_461_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_462_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_463_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_464_order__antisym,axiom,
! [X: nat > real,Y: nat > real] :
( ( ord_less_eq_nat_real @ X @ Y )
=> ( ( ord_less_eq_nat_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_465_order__antisym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_466_order__antisym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_467_order__antisym,axiom,
! [X: set_Ri1641125681238393385ccount,Y: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ X @ Y )
=> ( ( ord_le4487465848215339657ccount @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_468_order__antisym,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_469_order_Otrans,axiom,
! [A4: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_470_order_Otrans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ord_le4245800335709223507ccount @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_471_order_Otrans,axiom,
! [A4: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_472_order_Otrans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_473_order_Otrans,axiom,
! [A4: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ C2 )
=> ( ord_less_eq_int @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_474_order_Otrans,axiom,
! [A4: nat > real,B4: nat > real,C2: nat > real] :
( ( ord_less_eq_nat_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat_real @ B4 @ C2 )
=> ( ord_less_eq_nat_real @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_475_order_Otrans,axiom,
! [A4: set_int,B4: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ C2 )
=> ( ord_less_eq_set_int @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_476_order_Otrans,axiom,
! [A4: set_set_nat,B4: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_477_order_Otrans,axiom,
! [A4: set_Ri1641125681238393385ccount,B4: set_Ri1641125681238393385ccount,C2: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A4 @ B4 )
=> ( ( ord_le4487465848215339657ccount @ B4 @ C2 )
=> ( ord_le4487465848215339657ccount @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_478_order_Otrans,axiom,
! [A4: set_real,B4: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( ord_less_eq_set_real @ B4 @ C2 )
=> ( ord_less_eq_set_real @ A4 @ C2 ) ) ) ).
% order.trans
thf(fact_479_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_480_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z2 )
=> ( ord_le4245800335709223507ccount @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_481_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_482_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_483_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_484_order__trans,axiom,
! [X: nat > real,Y: nat > real,Z2: nat > real] :
( ( ord_less_eq_nat_real @ X @ Y )
=> ( ( ord_less_eq_nat_real @ Y @ Z2 )
=> ( ord_less_eq_nat_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_485_order__trans,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z2 )
=> ( ord_less_eq_set_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_486_order__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_487_order__trans,axiom,
! [X: set_Ri1641125681238393385ccount,Y: set_Ri1641125681238393385ccount,Z2: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ X @ Y )
=> ( ( ord_le4487465848215339657ccount @ Y @ Z2 )
=> ( ord_le4487465848215339657ccount @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_488_order__trans,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ Z2 )
=> ( ord_less_eq_set_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_489_linorder__wlog,axiom,
! [P: real > real > $o,A4: real,B4: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_490_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B4: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_491_linorder__wlog,axiom,
! [P: int > int > $o,A4: int,B4: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_492_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_493_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A5: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B5 @ A5 )
& ( ord_le4245800335709223507ccount @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_494_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_495_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_496_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_497_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat > real,Z: nat > real] : ( Y2 = Z ) )
= ( ^ [A5: nat > real,B5: nat > real] :
( ( ord_less_eq_nat_real @ B5 @ A5 )
& ( ord_less_eq_nat_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_498_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A5: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ B5 @ A5 )
& ( ord_less_eq_set_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_499_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A5 )
& ( ord_le6893508408891458716et_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_500_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Ri1641125681238393385ccount,Z: set_Ri1641125681238393385ccount] : ( Y2 = Z ) )
= ( ^ [A5: set_Ri1641125681238393385ccount,B5: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ B5 @ A5 )
& ( ord_le4487465848215339657ccount @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_501_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ A5 )
& ( ord_less_eq_set_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_502_dual__order_Oantisym,axiom,
! [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_503_dual__order_Oantisym,axiom,
! [B4: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_504_dual__order_Oantisym,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_505_dual__order_Oantisym,axiom,
! [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_506_dual__order_Oantisym,axiom,
! [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_507_dual__order_Oantisym,axiom,
! [B4: nat > real,A4: nat > real] :
( ( ord_less_eq_nat_real @ B4 @ A4 )
=> ( ( ord_less_eq_nat_real @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_508_dual__order_Oantisym,axiom,
! [B4: set_int,A4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_509_dual__order_Oantisym,axiom,
! [B4: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_510_dual__order_Oantisym,axiom,
! [B4: set_Ri1641125681238393385ccount,A4: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ B4 @ A4 )
=> ( ( ord_le4487465848215339657ccount @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_511_dual__order_Oantisym,axiom,
! [B4: set_real,A4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_512_dual__order_Otrans,axiom,
! [B4: real,A4: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_eq_real @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_513_dual__order_Otrans,axiom,
! [B4: risk_Free_account,A4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C2 @ B4 )
=> ( ord_le4245800335709223507ccount @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_514_dual__order_Otrans,axiom,
! [B4: nat,A4: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_515_dual__order_Otrans,axiom,
! [B4: set_nat,A4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( ord_less_eq_set_nat @ C2 @ B4 )
=> ( ord_less_eq_set_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_516_dual__order_Otrans,axiom,
! [B4: int,A4: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ C2 @ B4 )
=> ( ord_less_eq_int @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_517_dual__order_Otrans,axiom,
! [B4: nat > real,A4: nat > real,C2: nat > real] :
( ( ord_less_eq_nat_real @ B4 @ A4 )
=> ( ( ord_less_eq_nat_real @ C2 @ B4 )
=> ( ord_less_eq_nat_real @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_518_dual__order_Otrans,axiom,
! [B4: set_int,A4: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( ( ord_less_eq_set_int @ C2 @ B4 )
=> ( ord_less_eq_set_int @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_519_dual__order_Otrans,axiom,
! [B4: set_set_nat,A4: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B4 )
=> ( ord_le6893508408891458716et_nat @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_520_dual__order_Otrans,axiom,
! [B4: set_Ri1641125681238393385ccount,A4: set_Ri1641125681238393385ccount,C2: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ B4 @ A4 )
=> ( ( ord_le4487465848215339657ccount @ C2 @ B4 )
=> ( ord_le4487465848215339657ccount @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_521_dual__order_Otrans,axiom,
! [B4: set_real,A4: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( ( ord_less_eq_set_real @ C2 @ B4 )
=> ( ord_less_eq_set_real @ C2 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_522_antisym,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_523_antisym,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_524_antisym,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_525_antisym,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_526_antisym,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_527_antisym,axiom,
! [A4: nat > real,B4: nat > real] :
( ( ord_less_eq_nat_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat_real @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_528_antisym,axiom,
! [A4: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_529_antisym,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_530_antisym,axiom,
! [A4: set_Ri1641125681238393385ccount,B4: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A4 @ B4 )
=> ( ( ord_le4487465848215339657ccount @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_531_antisym,axiom,
! [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_532_le__funD,axiom,
! [F: nat > real,G: nat > real,X: nat] :
( ( ord_less_eq_nat_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_533_le__funE,axiom,
! [F: nat > real,G: nat > real,X: nat] :
( ( ord_less_eq_nat_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_534_le__funI,axiom,
! [F: nat > real,G: nat > real] :
( ! [X2: nat] : ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_nat_real @ F @ G ) ) ).
% le_funI
thf(fact_535_le__fun__def,axiom,
( ord_less_eq_nat_real
= ( ^ [F2: nat > real,G2: nat > real] :
! [X3: nat] : ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_536_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_537_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A5: risk_Free_account,B5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A5 @ B5 )
& ( ord_le4245800335709223507ccount @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_538_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_539_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_540_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_541_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat > real,Z: nat > real] : ( Y2 = Z ) )
= ( ^ [A5: nat > real,B5: nat > real] :
( ( ord_less_eq_nat_real @ A5 @ B5 )
& ( ord_less_eq_nat_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_542_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A5: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A5 @ B5 )
& ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_543_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_nat,Z: set_set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_544_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Ri1641125681238393385ccount,Z: set_Ri1641125681238393385ccount] : ( Y2 = Z ) )
= ( ^ [A5: set_Ri1641125681238393385ccount,B5: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ A5 @ B5 )
& ( ord_le4487465848215339657ccount @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_545_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A5 @ B5 )
& ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_546_order__subst1,axiom,
! [A4: real,F: real > real,B4: real,C2: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_547_order__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_548_order__subst1,axiom,
! [A4: real,F: nat > real,B4: nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_549_order__subst1,axiom,
! [A4: real,F: int > real,B4: int,C2: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_550_order__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_551_order__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_552_order__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_553_order__subst1,axiom,
! [A4: risk_Free_account,F: int > risk_Free_account,B4: int,C2: int] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_554_order__subst1,axiom,
! [A4: nat,F: real > nat,B4: real,C2: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_555_order__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_556_order__subst2,axiom,
! [A4: real,B4: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_557_order__subst2,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_558_order__subst2,axiom,
! [A4: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_559_order__subst2,axiom,
! [A4: real,B4: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F @ B4 ) @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_560_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_561_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_562_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_563_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > int,C2: int] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F @ B4 ) @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_564_order__subst2,axiom,
! [A4: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_565_order__subst2,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_566_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_567_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_568_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_569_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_570_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_571_order__eq__refl,axiom,
! [X: nat > real,Y: nat > real] :
( ( X = Y )
=> ( ord_less_eq_nat_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_572_order__eq__refl,axiom,
! [X: set_int,Y: set_int] :
( ( X = Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_573_order__eq__refl,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( X = Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_574_order__eq__refl,axiom,
! [X: set_Ri1641125681238393385ccount,Y: set_Ri1641125681238393385ccount] :
( ( X = Y )
=> ( ord_le4487465848215339657ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_575_order__eq__refl,axiom,
! [X: set_real,Y: set_real] :
( ( X = Y )
=> ( ord_less_eq_set_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_576_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_577_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_578_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_579_ord__eq__le__subst,axiom,
! [A4: real,F: real > real,B4: real,C2: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_580_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C2: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_581_ord__eq__le__subst,axiom,
! [A4: nat,F: real > nat,B4: real,C2: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_582_ord__eq__le__subst,axiom,
! [A4: int,F: real > int,B4: real,C2: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_583_ord__eq__le__subst,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_584_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_585_ord__eq__le__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_586_ord__eq__le__subst,axiom,
! [A4: int,F: risk_Free_account > int,B4: risk_Free_account,C2: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_587_ord__eq__le__subst,axiom,
! [A4: real,F: nat > real,B4: nat,C2: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_588_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C2: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_589_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_590_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_591_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_592_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_593_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C2: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_594_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_595_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C2: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_596_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > int,C2: int] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_597_ord__le__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_598_ord__le__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C2: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_599_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_600_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_601_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_602_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_603_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_604_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_605_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_606_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_607_order__antisym__conv,axiom,
! [Y: nat > real,X: nat > real] :
( ( ord_less_eq_nat_real @ Y @ X )
=> ( ( ord_less_eq_nat_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_608_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_609_order__antisym__conv,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_610_order__antisym__conv,axiom,
! [Y: set_Ri1641125681238393385ccount,X: set_Ri1641125681238393385ccount] :
( ( ord_le4487465848215339657ccount @ Y @ X )
=> ( ( ord_le4487465848215339657ccount @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_611_order__antisym__conv,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( ord_less_eq_set_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_612_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_613_top__empty__eq,axiom,
( top_to1164116034759999212ount_o
= ( ^ [X3: risk_Free_account] : ( member5612106785598075018ccount @ X3 @ top_to4387612366039908569ccount ) ) ) ).
% top_empty_eq
thf(fact_614_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_615_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_616_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_617_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_618_top__empty__eq,axiom,
( top_to5616354022731438592unit_o
= ( ^ [X3: set_Product_unit] : ( member5877623283571906838t_unit @ X3 @ top_to1767297665138865437t_unit ) ) ) ).
% top_empty_eq
thf(fact_619_top__empty__eq,axiom,
( top_top_set_int_o
= ( ^ [X3: set_int] : ( member_set_int @ X3 @ top_top_set_set_int ) ) ) ).
% top_empty_eq
thf(fact_620_top__empty__eq,axiom,
( top_top_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ) ) ).
% top_empty_eq
thf(fact_621_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_622_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_623_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_624_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_625_subset__Collect__iff,axiom,
! [B: set_nat_real,A: set_nat_real,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B @ A )
=> ( ( ord_le2908806416726583473t_real @ B
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_626_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_627_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_628_subset__Collect__iff,axiom,
! [B: set_int,A: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ B
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_629_subset__Collect__iff,axiom,
! [B: set_set_nat,A: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ B
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_630_subset__Collect__iff,axiom,
! [B: set_Ri1641125681238393385ccount,A: set_Ri1641125681238393385ccount,P: risk_Free_account > $o] :
( ( ord_le4487465848215339657ccount @ B @ A )
=> ( ( ord_le4487465848215339657ccount @ B
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_631_subset__Collect__iff,axiom,
! [B: set_real,A: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( ord_less_eq_set_real @ B
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_632_subset__CollectI,axiom,
! [B: set_nat_real,A: set_nat_real,Q: ( nat > real ) > $o,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B @ A )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_633_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_634_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_635_subset__CollectI,axiom,
! [B: set_int,A: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_636_subset__CollectI,axiom,
! [B: set_set_nat,A: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_637_subset__CollectI,axiom,
! [B: set_Ri1641125681238393385ccount,A: set_Ri1641125681238393385ccount,Q: risk_Free_account > $o,P: risk_Free_account > $o] :
( ( ord_le4487465848215339657ccount @ B @ A )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le4487465848215339657ccount
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_638_subset__CollectI,axiom,
! [B: set_real,A: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_639_Collect__restrict,axiom,
! [X5: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_640_Collect__restrict,axiom,
! [X5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_641_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_642_Collect__restrict,axiom,
! [X5: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_643_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_644_Collect__restrict,axiom,
! [X5: set_Ri1641125681238393385ccount,P: risk_Free_account > $o] :
( ord_le4487465848215339657ccount
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_645_Collect__restrict,axiom,
! [X5: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_646_prop__restrict,axiom,
! [X: nat > real,Z3: set_nat_real,X5: set_nat_real,P: ( nat > real ) > $o] :
( ( member_nat_real @ X @ Z3 )
=> ( ( ord_le2908806416726583473t_real @ Z3
@ ( collect_nat_real
@ ^ [X3: nat > real] :
( ( member_nat_real @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_647_prop__restrict,axiom,
! [X: a,Z3: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_648_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_649_prop__restrict,axiom,
! [X: int,Z3: set_int,X5: set_int,P: int > $o] :
( ( member_int @ X @ Z3 )
=> ( ( ord_less_eq_set_int @ Z3
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_650_prop__restrict,axiom,
! [X: set_nat,Z3: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_651_prop__restrict,axiom,
! [X: risk_Free_account,Z3: set_Ri1641125681238393385ccount,X5: set_Ri1641125681238393385ccount,P: risk_Free_account > $o] :
( ( member5612106785598075018ccount @ X @ Z3 )
=> ( ( ord_le4487465848215339657ccount @ Z3
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_652_prop__restrict,axiom,
! [X: real,Z3: set_real,X5: set_real,P: real > $o] :
( ( member_real @ X @ Z3 )
=> ( ( ord_less_eq_set_real @ Z3
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_653_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_654_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_655_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_656_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_657_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_658_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_659_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_660_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_661_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_662_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_663_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_664_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_665_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_666_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_667_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_668_Abs__account__induct,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( P @ ( risk_F5458100604530014700ccount @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_account_induct
thf(fact_669_Abs__account__cases,axiom,
! [X: risk_Free_account] :
~ ! [Y3: nat > real] :
( ( X
= ( risk_F5458100604530014700ccount @ Y3 ) )
=> ~ ( member_nat_real @ Y3 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ) ).
% Abs_account_cases
thf(fact_670_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_671_sum__abs__ge__zero,axiom,
! [F: nat > real,A: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs_ge_zero
thf(fact_672_sum__abs__ge__zero,axiom,
! [F: real > int,A: set_real] :
( ord_less_eq_int @ zero_zero_int
@ ( groups1932886352136224148al_int
@ ^ [I2: real] : ( abs_abs_int @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs_ge_zero
thf(fact_673_sum__abs__ge__zero,axiom,
! [F: ( nat > real ) > real,A: set_nat_real] :
( ord_less_eq_real @ zero_zero_real
@ ( groups4253619806861319043l_real
@ ^ [I2: nat > real] : ( abs_abs_real @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs_ge_zero
thf(fact_674_sum__abs__ge__zero,axiom,
! [F: ( nat > real ) > int,A: set_nat_real] :
( ord_less_eq_int @ zero_zero_int
@ ( groups777517501785750147al_int
@ ^ [I2: nat > real] : ( abs_abs_int @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs_ge_zero
thf(fact_675_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_676_Greatest__equality,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ( P @ X )
=> ( ! [Y3: risk_Free_account] :
( ( P @ Y3 )
=> ( ord_le4245800335709223507ccount @ Y3 @ X ) )
=> ( ( order_4130363404468270554ccount @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_677_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_678_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_679_Greatest__equality,axiom,
! [P: ( nat > real ) > $o,X: nat > real] :
( ( P @ X )
=> ( ! [Y3: nat > real] :
( ( P @ Y3 )
=> ( ord_less_eq_nat_real @ Y3 @ X ) )
=> ( ( order_1398359769023000578t_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_680_Greatest__equality,axiom,
! [P: set_int > $o,X: set_int] :
( ( P @ X )
=> ( ! [Y3: set_int] :
( ( P @ Y3 )
=> ( ord_less_eq_set_int @ Y3 @ X ) )
=> ( ( order_1546957118920008137et_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_681_Greatest__equality,axiom,
! [P: set_set_nat > $o,X: set_set_nat] :
( ( P @ X )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X ) )
=> ( ( order_1279421399067128355et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_682_Greatest__equality,axiom,
! [P: set_Ri1641125681238393385ccount > $o,X: set_Ri1641125681238393385ccount] :
( ( P @ X )
=> ( ! [Y3: set_Ri1641125681238393385ccount] :
( ( P @ Y3 )
=> ( ord_le4487465848215339657ccount @ Y3 @ X ) )
=> ( ( order_1250964040836243984ccount @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_683_Greatest__equality,axiom,
! [P: set_real > $o,X: set_real] :
( ( P @ X )
=> ( ! [Y3: set_real] :
( ( P @ Y3 )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ( order_1598108641013654857t_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_684_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_685_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ! [X2: real] :
( ( P @ X2 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_686_GreatestI2__order,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account,Q: risk_Free_account > $o] :
( ( P @ X )
=> ( ! [Y3: risk_Free_account] :
( ( P @ Y3 )
=> ( ord_le4245800335709223507ccount @ Y3 @ X ) )
=> ( ! [X2: risk_Free_account] :
( ( P @ X2 )
=> ( ! [Y5: risk_Free_account] :
( ( P @ Y5 )
=> ( ord_le4245800335709223507ccount @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_4130363404468270554ccount @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_687_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_688_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_689_GreatestI2__order,axiom,
! [P: ( nat > real ) > $o,X: nat > real,Q: ( nat > real ) > $o] :
( ( P @ X )
=> ( ! [Y3: nat > real] :
( ( P @ Y3 )
=> ( ord_less_eq_nat_real @ Y3 @ X ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ( ! [Y5: nat > real] :
( ( P @ Y5 )
=> ( ord_less_eq_nat_real @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1398359769023000578t_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_690_GreatestI2__order,axiom,
! [P: set_int > $o,X: set_int,Q: set_int > $o] :
( ( P @ X )
=> ( ! [Y3: set_int] :
( ( P @ Y3 )
=> ( ord_less_eq_set_int @ Y3 @ X ) )
=> ( ! [X2: set_int] :
( ( P @ X2 )
=> ( ! [Y5: set_int] :
( ( P @ Y5 )
=> ( ord_less_eq_set_int @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1546957118920008137et_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_691_GreatestI2__order,axiom,
! [P: set_set_nat > $o,X: set_set_nat,Q: set_set_nat > $o] :
( ( P @ X )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X ) )
=> ( ! [X2: set_set_nat] :
( ( P @ X2 )
=> ( ! [Y5: set_set_nat] :
( ( P @ Y5 )
=> ( ord_le6893508408891458716et_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_692_GreatestI2__order,axiom,
! [P: set_Ri1641125681238393385ccount > $o,X: set_Ri1641125681238393385ccount,Q: set_Ri1641125681238393385ccount > $o] :
( ( P @ X )
=> ( ! [Y3: set_Ri1641125681238393385ccount] :
( ( P @ Y3 )
=> ( ord_le4487465848215339657ccount @ Y3 @ X ) )
=> ( ! [X2: set_Ri1641125681238393385ccount] :
( ( P @ X2 )
=> ( ! [Y5: set_Ri1641125681238393385ccount] :
( ( P @ Y5 )
=> ( ord_le4487465848215339657ccount @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1250964040836243984ccount @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_693_GreatestI2__order,axiom,
! [P: set_real > $o,X: set_real,Q: set_real > $o] :
( ( P @ X )
=> ( ! [Y3: set_real] :
( ( P @ Y3 )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ! [X2: set_real] :
( ( P @ X2 )
=> ( ! [Y5: set_real] :
( ( P @ Y5 )
=> ( ord_less_eq_set_real @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1598108641013654857t_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_694_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_695_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_696_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_697_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_698_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > real,B: real,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= B )
=> ( ( member_real @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_699_sum__nonneg__leq__bound,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > real,B: real,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8212752698507433307t_real @ F @ S2 )
= B )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_700_sum__nonneg__leq__bound,axiom,
! [S2: set_a,F: a > real,B: real,I: a] :
( ( finite_finite_a @ S2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups2740460157737275248a_real @ F @ S2 )
= B )
=> ( ( member_a @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_701_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > real,B: real,I: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= B )
=> ( ( member_int @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_702_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > risk_Free_account,B: risk_Free_account,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups8516999891779824987ccount @ F @ S2 )
= B )
=> ( ( member_real @ I @ S2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_703_sum__nonneg__leq__bound,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > risk_Free_account,B: risk_Free_account,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups5726723449141454370ccount @ F @ S2 )
= B )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_704_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > risk_Free_account,B: risk_Free_account,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups6033208628184776703ccount @ F @ S2 )
= B )
=> ( ( member_nat @ I @ S2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_705_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > risk_Free_account,B: risk_Free_account,I: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups2220918773033463387ccount @ F @ S2 )
= B )
=> ( ( member_int @ I @ S2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_706_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > nat,B: nat,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= B )
=> ( ( member_real @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_707_sum__nonneg__leq__bound,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > nat,B: nat,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1716945928530811391nt_nat @ F @ S2 )
= B )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_708_sum__nonneg__0,axiom,
! [S2: set_real,F: real > real,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_real @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_709_sum__nonneg__0,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > real,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8212752698507433307t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_710_sum__nonneg__0,axiom,
! [S2: set_a,F: a > real,I: a] :
( ( finite_finite_a @ S2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups2740460157737275248a_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_a @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_711_sum__nonneg__0,axiom,
! [S2: set_int,F: int > real,I: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_int @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_712_sum__nonneg__0,axiom,
! [S2: set_real,F: real > risk_Free_account,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups8516999891779824987ccount @ F @ S2 )
= zero_z1425366712893667068ccount )
=> ( ( member_real @ I @ S2 )
=> ( ( F @ I )
= zero_z1425366712893667068ccount ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_713_sum__nonneg__0,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > risk_Free_account,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups5726723449141454370ccount @ F @ S2 )
= zero_z1425366712893667068ccount )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ( F @ I )
= zero_z1425366712893667068ccount ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_714_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > risk_Free_account,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups6033208628184776703ccount @ F @ S2 )
= zero_z1425366712893667068ccount )
=> ( ( member_nat @ I @ S2 )
=> ( ( F @ I )
= zero_z1425366712893667068ccount ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_715_sum__nonneg__0,axiom,
! [S2: set_int,F: int > risk_Free_account,I: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ I3 ) ) )
=> ( ( ( groups2220918773033463387ccount @ F @ S2 )
= zero_z1425366712893667068ccount )
=> ( ( member_int @ I @ S2 )
=> ( ( F @ I )
= zero_z1425366712893667068ccount ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_716_sum__nonneg__0,axiom,
! [S2: set_real,F: real > nat,I: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_real @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_717_sum__nonneg__0,axiom,
! [S2: set_Ri1641125681238393385ccount,F: risk_Free_account > nat,I: risk_Free_account] :
( ( finite1362240334998357386ccount @ S2 )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1716945928530811391nt_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member5612106785598075018ccount @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_718_Icc__subset__Iic__iff,axiom,
! [L2: risk_Free_account,H: risk_Free_account,H2: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or4484699493994522366ccount @ L2 @ H ) @ ( set_or3854930313887350124ccount @ H2 ) )
= ( ~ ( ord_le4245800335709223507ccount @ L2 @ H )
| ( ord_le4245800335709223507ccount @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_719_Icc__subset__Iic__iff,axiom,
! [L2: real,H: real,H2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L2 @ H ) @ ( set_ord_atMost_real @ H2 ) )
= ( ~ ( ord_less_eq_real @ L2 @ H )
| ( ord_less_eq_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_720_Icc__subset__Iic__iff,axiom,
! [L2: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_721_Icc__subset__Iic__iff,axiom,
! [L2: int,H: int,H2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L2 @ H ) @ ( set_ord_atMost_int @ H2 ) )
= ( ~ ( ord_less_eq_int @ L2 @ H )
| ( ord_less_eq_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_722_Icc__subset__Iic__iff,axiom,
! [L2: set_Ri1641125681238393385ccount,H: set_Ri1641125681238393385ccount,H2: set_Ri1641125681238393385ccount] :
( ( ord_le5934089292365423551ccount @ ( set_or8103168004485710644ccount @ L2 @ H ) @ ( set_or2377417256128143778ccount @ H2 ) )
= ( ~ ( ord_le4487465848215339657ccount @ L2 @ H )
| ( ord_le4487465848215339657ccount @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_723_Icc__subset__Iic__iff,axiom,
! [L2: set_real,H: set_real,H2: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ L2 @ H ) @ ( set_or5092868708245317595t_real @ H2 ) )
= ( ~ ( ord_less_eq_set_real @ L2 @ H )
| ( ord_less_eq_set_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_724_Icc__subset__Iic__iff,axiom,
! [L2: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L2 @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_725_Icc__subset__Iic__iff,axiom,
! [L2: set_int,H: set_int,H2: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L2 @ H ) @ ( set_or58775011639299419et_int @ H2 ) )
= ( ~ ( ord_less_eq_set_int @ L2 @ H )
| ( ord_less_eq_set_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_726_Icc__subset__Iic__iff,axiom,
! [L2: set_a,H: set_a,H2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L2 @ H ) @ ( set_ord_atMost_set_a @ H2 ) )
= ( ~ ( ord_less_eq_set_a @ L2 @ H )
| ( ord_less_eq_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_727_Icc__subset__Iic__iff,axiom,
! [L2: set_set_nat,H: set_set_nat,H2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ L2 @ H ) @ ( set_or7210490968680142261et_nat @ H2 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ L2 @ H )
| ( ord_le6893508408891458716et_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_728_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_729_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_730_abs__idempotent,axiom,
! [A4: real] :
( ( abs_abs_real @ ( abs_abs_real @ A4 ) )
= ( abs_abs_real @ A4 ) ) ).
% abs_idempotent
thf(fact_731_abs__idempotent,axiom,
! [A4: int] :
( ( abs_abs_int @ ( abs_abs_int @ A4 ) )
= ( abs_abs_int @ A4 ) ) ).
% abs_idempotent
thf(fact_732_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_733_abs__0__eq,axiom,
! [A4: real] :
( ( zero_zero_real
= ( abs_abs_real @ A4 ) )
= ( A4 = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_734_abs__0__eq,axiom,
! [A4: int] :
( ( zero_zero_int
= ( abs_abs_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_735_abs__eq__0,axiom,
! [A4: real] :
( ( ( abs_abs_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_736_abs__eq__0,axiom,
! [A4: int] :
( ( ( abs_abs_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_737_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_738_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_739_Icc__eq__Icc,axiom,
! [L2: real,H: real,L3: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L2 @ H )
= ( set_or1222579329274155063t_real @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L2 @ H )
& ~ ( ord_less_eq_real @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_740_Icc__eq__Icc,axiom,
! [L2: risk_Free_account,H: risk_Free_account,L3: risk_Free_account,H2: risk_Free_account] :
( ( ( set_or4484699493994522366ccount @ L2 @ H )
= ( set_or4484699493994522366ccount @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_le4245800335709223507ccount @ L2 @ H )
& ~ ( ord_le4245800335709223507ccount @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_741_Icc__eq__Icc,axiom,
! [L2: set_nat,H: set_nat,L3: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L2 @ H )
= ( set_or4548717258645045905et_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L2 @ H )
& ~ ( ord_less_eq_set_nat @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_742_Icc__eq__Icc,axiom,
! [L2: nat > real,H: nat > real,L3: nat > real,H2: nat > real] :
( ( ( set_or1752695858549252390t_real @ L2 @ H )
= ( set_or1752695858549252390t_real @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat_real @ L2 @ H )
& ~ ( ord_less_eq_nat_real @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_743_Icc__eq__Icc,axiom,
! [L2: set_int,H: set_int,L3: set_int,H2: set_int] :
( ( ( set_or370866239135849197et_int @ L2 @ H )
= ( set_or370866239135849197et_int @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_int @ L2 @ H )
& ~ ( ord_less_eq_set_int @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_744_Icc__eq__Icc,axiom,
! [L2: set_set_nat,H: set_set_nat,L3: set_set_nat,H2: set_set_nat] :
( ( ( set_or9137876137106135879et_nat @ L2 @ H )
= ( set_or9137876137106135879et_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_le6893508408891458716et_nat @ L2 @ H )
& ~ ( ord_le6893508408891458716et_nat @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_745_Icc__eq__Icc,axiom,
! [L2: set_Ri1641125681238393385ccount,H: set_Ri1641125681238393385ccount,L3: set_Ri1641125681238393385ccount,H2: set_Ri1641125681238393385ccount] :
( ( ( set_or8103168004485710644ccount @ L2 @ H )
= ( set_or8103168004485710644ccount @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_le4487465848215339657ccount @ L2 @ H )
& ~ ( ord_le4487465848215339657ccount @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_746_Icc__eq__Icc,axiom,
! [L2: set_real,H: set_real,L3: set_real,H2: set_real] :
( ( ( set_or7743017856606604397t_real @ L2 @ H )
= ( set_or7743017856606604397t_real @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_real @ L2 @ H )
& ~ ( ord_less_eq_set_real @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_747_Icc__eq__Icc,axiom,
! [L2: nat,H: nat,L3: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L2 @ H )
= ( set_or1269000886237332187st_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L2 @ H )
& ~ ( ord_less_eq_nat @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_748_Icc__eq__Icc,axiom,
! [L2: int,H: int,L3: int,H2: int] :
( ( ( set_or1266510415728281911st_int @ L2 @ H )
= ( set_or1266510415728281911st_int @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_int @ L2 @ H )
& ~ ( ord_less_eq_int @ L3 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_749_atLeastAtMost__iff,axiom,
! [I: real,L2: real,U2: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U2 ) )
= ( ( ord_less_eq_real @ L2 @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_750_atLeastAtMost__iff,axiom,
! [I: risk_Free_account,L2: risk_Free_account,U2: risk_Free_account] :
( ( member5612106785598075018ccount @ I @ ( set_or4484699493994522366ccount @ L2 @ U2 ) )
= ( ( ord_le4245800335709223507ccount @ L2 @ I )
& ( ord_le4245800335709223507ccount @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_751_atLeastAtMost__iff,axiom,
! [I: set_nat,L2: set_nat,U2: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_set_nat @ L2 @ I )
& ( ord_less_eq_set_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_752_atLeastAtMost__iff,axiom,
! [I: nat > real,L2: nat > real,U2: nat > real] :
( ( member_nat_real @ I @ ( set_or1752695858549252390t_real @ L2 @ U2 ) )
= ( ( ord_less_eq_nat_real @ L2 @ I )
& ( ord_less_eq_nat_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_753_atLeastAtMost__iff,axiom,
! [I: set_int,L2: set_int,U2: set_int] :
( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L2 @ U2 ) )
= ( ( ord_less_eq_set_int @ L2 @ I )
& ( ord_less_eq_set_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_754_atLeastAtMost__iff,axiom,
! [I: set_set_nat,L2: set_set_nat,U2: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or9137876137106135879et_nat @ L2 @ U2 ) )
= ( ( ord_le6893508408891458716et_nat @ L2 @ I )
& ( ord_le6893508408891458716et_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_755_atLeastAtMost__iff,axiom,
! [I: set_Ri1641125681238393385ccount,L2: set_Ri1641125681238393385ccount,U2: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ I @ ( set_or8103168004485710644ccount @ L2 @ U2 ) )
= ( ( ord_le4487465848215339657ccount @ L2 @ I )
& ( ord_le4487465848215339657ccount @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_756_atLeastAtMost__iff,axiom,
! [I: set_real,L2: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or7743017856606604397t_real @ L2 @ U2 ) )
= ( ( ord_less_eq_set_real @ L2 @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_757_atLeastAtMost__iff,axiom,
! [I: nat,L2: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_nat @ L2 @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_758_atLeastAtMost__iff,axiom,
! [I: int,L2: int,U2: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U2 ) )
= ( ( ord_less_eq_int @ L2 @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_759_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_760_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_761_abs__sum__abs,axiom,
! [F: nat > real,A: set_nat] :
( ( abs_abs_real
@ ( groups6591440286371151544t_real
@ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
@ A ) )
= ( groups6591440286371151544t_real
@ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
@ A ) ) ).
% abs_sum_abs
thf(fact_762_abs__sum__abs,axiom,
! [F: real > int,A: set_real] :
( ( abs_abs_int
@ ( groups1932886352136224148al_int
@ ^ [A5: real] : ( abs_abs_int @ ( F @ A5 ) )
@ A ) )
= ( groups1932886352136224148al_int
@ ^ [A5: real] : ( abs_abs_int @ ( F @ A5 ) )
@ A ) ) ).
% abs_sum_abs
thf(fact_763_abs__sum__abs,axiom,
! [F: ( nat > real ) > real,A: set_nat_real] :
( ( abs_abs_real
@ ( groups4253619806861319043l_real
@ ^ [A5: nat > real] : ( abs_abs_real @ ( F @ A5 ) )
@ A ) )
= ( groups4253619806861319043l_real
@ ^ [A5: nat > real] : ( abs_abs_real @ ( F @ A5 ) )
@ A ) ) ).
% abs_sum_abs
thf(fact_764_abs__sum__abs,axiom,
! [F: ( nat > real ) > int,A: set_nat_real] :
( ( abs_abs_int
@ ( groups777517501785750147al_int
@ ^ [A5: nat > real] : ( abs_abs_int @ ( F @ A5 ) )
@ A ) )
= ( groups777517501785750147al_int
@ ^ [A5: nat > real] : ( abs_abs_int @ ( F @ A5 ) )
@ A ) ) ).
% abs_sum_abs
thf(fact_765_abs__of__nonneg,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( abs_abs_real @ A4 )
= A4 ) ) ).
% abs_of_nonneg
thf(fact_766_abs__of__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( abs_abs_int @ A4 )
= A4 ) ) ).
% abs_of_nonneg
thf(fact_767_abs__le__self__iff,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ A4 )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% abs_le_self_iff
thf(fact_768_abs__le__self__iff,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% abs_le_self_iff
thf(fact_769_abs__le__zero__iff,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_770_abs__le__zero__iff,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_771_sum_Oinfinite,axiom,
! [A: set_int,G: int > real] :
( ~ ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_772_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > risk_Free_account] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_773_sum_Oinfinite,axiom,
! [A: set_int,G: int > risk_Free_account] :
( ~ ( finite_finite_int @ A )
=> ( ( groups2220918773033463387ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_774_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_775_sum_Oinfinite,axiom,
! [A: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_776_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_777_sum_Oinfinite,axiom,
! [A: set_int,G: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_778_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_779_sum_Oinfinite,axiom,
! [A: set_a,G: a > risk_Free_account] :
( ~ ( finite_finite_a @ A )
=> ( ( groups4655409347963886775ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_780_sum_Oinfinite,axiom,
! [A: set_real,G: real > int] :
( ~ ( finite_finite_real @ A )
=> ( ( groups1932886352136224148al_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_781_sum__eq__0__iff,axiom,
! [F3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F3 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_782_sum__eq__0__iff,axiom,
! [F3: set_int,F: int > nat] :
( ( finite_finite_int @ F3 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ F3 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_783_sum__eq__0__iff,axiom,
! [F3: set_nat_real,F: ( nat > real ) > nat] :
( ( finite7853608736407863218t_real @ F3 )
=> ( ( ( groups780007972294800423al_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ F3 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_784_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_785_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_786_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_787_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_788_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_789_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_790_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_791_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_792_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_793_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_794_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_795_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_796_atLeastatMost__subset__iff,axiom,
! [A4: nat > real,B4: nat > real,C2: nat > real,D: nat > real] :
( ( ord_le2908806416726583473t_real @ ( set_or1752695858549252390t_real @ A4 @ B4 ) @ ( set_or1752695858549252390t_real @ C2 @ D ) )
= ( ~ ( ord_less_eq_nat_real @ A4 @ B4 )
| ( ( ord_less_eq_nat_real @ C2 @ A4 )
& ( ord_less_eq_nat_real @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_797_atLeastatMost__subset__iff,axiom,
! [A4: set_int,B4: set_int,C2: set_int,D: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A4 @ B4 ) @ ( set_or370866239135849197et_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_set_int @ A4 @ B4 )
| ( ( ord_less_eq_set_int @ C2 @ A4 )
& ( ord_less_eq_set_int @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_798_atLeastatMost__subset__iff,axiom,
! [A4: set_set_nat,B4: set_set_nat,C2: set_set_nat,D: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ A4 @ B4 ) @ ( set_or9137876137106135879et_nat @ C2 @ D ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A4 @ B4 )
| ( ( ord_le6893508408891458716et_nat @ C2 @ A4 )
& ( ord_le6893508408891458716et_nat @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_799_atLeastatMost__subset__iff,axiom,
! [A4: set_Ri1641125681238393385ccount,B4: set_Ri1641125681238393385ccount,C2: set_Ri1641125681238393385ccount,D: set_Ri1641125681238393385ccount] :
( ( ord_le5934089292365423551ccount @ ( set_or8103168004485710644ccount @ A4 @ B4 ) @ ( set_or8103168004485710644ccount @ C2 @ D ) )
= ( ~ ( ord_le4487465848215339657ccount @ A4 @ B4 )
| ( ( ord_le4487465848215339657ccount @ C2 @ A4 )
& ( ord_le4487465848215339657ccount @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_800_atLeastatMost__subset__iff,axiom,
! [A4: set_real,B4: set_real,C2: set_real,D: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A4 @ B4 ) @ ( set_or7743017856606604397t_real @ C2 @ D ) )
= ( ~ ( ord_less_eq_set_real @ A4 @ B4 )
| ( ( ord_less_eq_set_real @ C2 @ A4 )
& ( ord_less_eq_set_real @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_801_atLeastatMost__subset__iff,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A4 @ B4 ) @ ( set_or4548717258645045905et_nat @ C2 @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A4 @ B4 )
| ( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( ord_less_eq_set_nat @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_802_atLeastatMost__subset__iff,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C2: risk_Free_account,D: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or4484699493994522366ccount @ A4 @ B4 ) @ ( set_or4484699493994522366ccount @ C2 @ D ) )
= ( ~ ( ord_le4245800335709223507ccount @ A4 @ B4 )
| ( ( ord_le4245800335709223507ccount @ C2 @ A4 )
& ( ord_le4245800335709223507ccount @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_803_atLeastatMost__subset__iff,axiom,
! [A4: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A4 @ B4 ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
= ( ~ ( ord_less_eq_real @ A4 @ B4 )
| ( ( ord_less_eq_real @ C2 @ A4 )
& ( ord_less_eq_real @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_804_atLeastatMost__subset__iff,axiom,
! [A4: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B4 )
| ( ( ord_less_eq_nat @ C2 @ A4 )
& ( ord_less_eq_nat @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_805_atLeastatMost__subset__iff,axiom,
! [A4: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A4 @ B4 ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_int @ A4 @ B4 )
| ( ( ord_less_eq_int @ C2 @ A4 )
& ( ord_less_eq_int @ B4 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_806_sum_Odelta,axiom,
! [S: set_real,A4: real,B4: real > real] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_807_sum_Odelta,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > real] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups8212752698507433307t_real
@ ^ [K3: risk_Free_account] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups8212752698507433307t_real
@ ^ [K3: risk_Free_account] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_808_sum_Odelta,axiom,
! [S: set_a,A4: a,B4: a > real] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A4 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K3: a] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_a @ A4 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K3: a] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_809_sum_Odelta,axiom,
! [S: set_int,A4: int,B4: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A4 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_int @ A4 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_810_sum_Odelta,axiom,
! [S: set_real,A4: real,B4: real > risk_Free_account] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_811_sum_Odelta,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > risk_Free_account] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups5726723449141454370ccount
@ ^ [K3: risk_Free_account] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups5726723449141454370ccount
@ ^ [K3: risk_Free_account] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_812_sum_Odelta,axiom,
! [S: set_nat,A4: nat,B4: nat > risk_Free_account] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A4 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_nat @ A4 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_813_sum_Odelta,axiom,
! [S: set_int,A4: int,B4: int > risk_Free_account] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A4 @ S )
=> ( ( groups2220918773033463387ccount
@ ^ [K3: int] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_int @ A4 @ S )
=> ( ( groups2220918773033463387ccount
@ ^ [K3: int] : ( if_Risk_Free_account @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_814_sum_Odelta,axiom,
! [S: set_real,A4: real,B4: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_815_sum_Odelta,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > nat] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups1716945928530811391nt_nat
@ ^ [K3: risk_Free_account] : ( if_nat @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups1716945928530811391nt_nat
@ ^ [K3: risk_Free_account] : ( if_nat @ ( K3 = A4 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_816_sum_Odelta_H,axiom,
! [S: set_real,A4: real,B4: real > real] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_817_sum_Odelta_H,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > real] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups8212752698507433307t_real
@ ^ [K3: risk_Free_account] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups8212752698507433307t_real
@ ^ [K3: risk_Free_account] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_818_sum_Odelta_H,axiom,
! [S: set_a,A4: a,B4: a > real] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A4 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K3: a] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_a @ A4 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K3: a] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_819_sum_Odelta_H,axiom,
! [S: set_int,A4: int,B4: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A4 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_int @ A4 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_820_sum_Odelta_H,axiom,
! [S: set_real,A4: real,B4: real > risk_Free_account] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_821_sum_Odelta_H,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > risk_Free_account] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups5726723449141454370ccount
@ ^ [K3: risk_Free_account] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups5726723449141454370ccount
@ ^ [K3: risk_Free_account] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_822_sum_Odelta_H,axiom,
! [S: set_nat,A4: nat,B4: nat > risk_Free_account] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A4 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_nat @ A4 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_823_sum_Odelta_H,axiom,
! [S: set_int,A4: int,B4: int > risk_Free_account] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A4 @ S )
=> ( ( groups2220918773033463387ccount
@ ^ [K3: int] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_int @ A4 @ S )
=> ( ( groups2220918773033463387ccount
@ ^ [K3: int] : ( if_Risk_Free_account @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_824_sum_Odelta_H,axiom,
! [S: set_real,A4: real,B4: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A4 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member_real @ A4 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_825_sum_Odelta_H,axiom,
! [S: set_Ri1641125681238393385ccount,A4: risk_Free_account,B4: risk_Free_account > nat] :
( ( finite1362240334998357386ccount @ S )
=> ( ( ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups1716945928530811391nt_nat
@ ^ [K3: risk_Free_account] : ( if_nat @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= ( B4 @ A4 ) ) )
& ( ~ ( member5612106785598075018ccount @ A4 @ S )
=> ( ( groups1716945928530811391nt_nat
@ ^ [K3: risk_Free_account] : ( if_nat @ ( A4 = K3 ) @ ( B4 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_826_sum__abs,axiom,
! [F: nat > real,A: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A ) )
@ ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs
thf(fact_827_sum__abs,axiom,
! [F: real > int,A: set_real] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups1932886352136224148al_int @ F @ A ) )
@ ( groups1932886352136224148al_int
@ ^ [I2: real] : ( abs_abs_int @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs
thf(fact_828_sum__abs,axiom,
! [F: ( nat > real ) > real,A: set_nat_real] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups4253619806861319043l_real @ F @ A ) )
@ ( groups4253619806861319043l_real
@ ^ [I2: nat > real] : ( abs_abs_real @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs
thf(fact_829_sum__abs,axiom,
! [F: ( nat > real ) > int,A: set_nat_real] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups777517501785750147al_int @ F @ A ) )
@ ( groups777517501785750147al_int
@ ^ [I2: nat > real] : ( abs_abs_int @ ( F @ I2 ) )
@ A ) ) ).
% sum_abs
thf(fact_830_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_831_abs__le__D1,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ B4 )
=> ( ord_less_eq_real @ A4 @ B4 ) ) ).
% abs_le_D1
thf(fact_832_abs__le__D1,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ B4 )
=> ( ord_less_eq_int @ A4 @ B4 ) ) ).
% abs_le_D1
thf(fact_833_abs__ge__self,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ ( abs_abs_real @ A4 ) ) ).
% abs_ge_self
thf(fact_834_abs__ge__self,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ ( abs_abs_int @ A4 ) ) ).
% abs_ge_self
thf(fact_835_not__UNIV__eq__Icc,axiom,
! [L3: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L3 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_836_not__UNIV__eq__Icc,axiom,
! [L3: int,H2: int] :
( top_top_set_int
!= ( set_or1266510415728281911st_int @ L3 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_837_not__Iic__eq__Icc,axiom,
! [H2: int,L2: int,H: int] :
( ( set_ord_atMost_int @ H2 )
!= ( set_or1266510415728281911st_int @ L2 @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_838_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_839_type__copy__ex__RepI,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,F3: ( nat > real ) > $o] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ( ( ? [X6: nat > real] : ( F3 @ X6 ) )
= ( ? [B5: risk_Free_account] : ( F3 @ ( Rep @ B5 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_840_type__copy__obj__one__point__absE,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,S2: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ~ ! [X2: nat > real] :
( S2
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_841_infinite__Iic,axiom,
! [A4: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A4 ) ) ).
% infinite_Iic
thf(fact_842_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_843_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_nat,G: real > nat > real,R: real > nat > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8097168146408367636l_real
@ ^ [X3: real] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8097168146408367636l_real
@ ^ [X3: real] : ( G @ X3 @ Y4 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_844_sum_Oswap__restrict,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_nat,G: risk_Free_account > nat > real,R: risk_Free_account > nat > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8212752698507433307t_real
@ ^ [X3: risk_Free_account] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8212752698507433307t_real
@ ^ [X3: risk_Free_account] : ( G @ X3 @ Y4 )
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_845_sum_Oswap__restrict,axiom,
! [A: set_a,B: set_nat,G: a > nat > real,R: a > nat > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups2740460157737275248a_real
@ ^ [X3: a] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups2740460157737275248a_real
@ ^ [X3: a] : ( G @ X3 @ Y4 )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_846_sum_Oswap__restrict,axiom,
! [A: set_int,B: set_nat,G: int > nat > real,R: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8778361861064173332t_real
@ ^ [X3: int] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8778361861064173332t_real
@ ^ [X3: int] : ( G @ X3 @ Y4 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_847_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_a,G: real > a > risk_Free_account,R: real > a > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_a @ B )
=> ( ( groups8516999891779824987ccount
@ ^ [X3: real] :
( groups4655409347963886775ccount @ ( G @ X3 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups8516999891779824987ccount
@ ^ [X3: real] : ( G @ X3 @ Y4 )
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_848_sum_Oswap__restrict,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_a,G: risk_Free_account > a > risk_Free_account,R: risk_Free_account > a > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( finite_finite_a @ B )
=> ( ( groups5726723449141454370ccount
@ ^ [X3: risk_Free_account] :
( groups4655409347963886775ccount @ ( G @ X3 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups5726723449141454370ccount
@ ^ [X3: risk_Free_account] : ( G @ X3 @ Y4 )
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_849_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_a,G: nat > a > risk_Free_account,R: nat > a > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ( groups6033208628184776703ccount
@ ^ [X3: nat] :
( groups4655409347963886775ccount @ ( G @ X3 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups6033208628184776703ccount
@ ^ [X3: nat] : ( G @ X3 @ Y4 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_850_sum_Oswap__restrict,axiom,
! [A: set_int,B: set_a,G: int > a > risk_Free_account,R: int > a > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B )
=> ( ( groups2220918773033463387ccount
@ ^ [X3: int] :
( groups4655409347963886775ccount @ ( G @ X3 )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups4655409347963886775ccount
@ ^ [Y4: a] :
( groups2220918773033463387ccount
@ ^ [X3: int] : ( G @ X3 @ Y4 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_851_sum_Oswap__restrict,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_real,G: risk_Free_account > real > int,R: risk_Free_account > real > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups1714455458021761115nt_int
@ ^ [X3: risk_Free_account] :
( groups1932886352136224148al_int @ ( G @ X3 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups1932886352136224148al_int
@ ^ [Y4: real] :
( groups1714455458021761115nt_int
@ ^ [X3: risk_Free_account] : ( G @ X3 @ Y4 )
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_852_sum_Oswap__restrict,axiom,
! [A: set_a,B: set_real,G: a > real > int,R: a > real > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups6332066207828071664_a_int
@ ^ [X3: a] :
( groups1932886352136224148al_int @ ( G @ X3 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B )
& ( R @ X3 @ Y4 ) ) ) )
@ A )
= ( groups1932886352136224148al_int
@ ^ [Y4: real] :
( groups6332066207828071664_a_int
@ ^ [X3: a] : ( G @ X3 @ Y4 )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( R @ X3 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_853_abs__ge__zero,axiom,
! [A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A4 ) ) ).
% abs_ge_zero
thf(fact_854_abs__ge__zero,axiom,
! [A4: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A4 ) ) ).
% abs_ge_zero
thf(fact_855_not__UNIV__le__Icc,axiom,
! [L2: real,H: real] :
~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_or1222579329274155063t_real @ L2 @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_856_not__UNIV__le__Icc,axiom,
! [L2: nat,H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_857_not__UNIV__le__Icc,axiom,
! [L2: int,H: int] :
~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_or1266510415728281911st_int @ L2 @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_858_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_859_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_860_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_861_not__Iic__le__Icc,axiom,
! [H: real,L3: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or1222579329274155063t_real @ L3 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_862_not__Iic__le__Icc,axiom,
! [H: int,L3: int,H2: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L3 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_863_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_864_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_865_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_866_sum__mono__inv,axiom,
! [F: real > real,I4: set_real,G: real > real,I: real] :
( ( ( groups8097168146408367636l_real @ F @ I4 )
= ( groups8097168146408367636l_real @ G @ I4 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I @ I4 )
=> ( ( finite_finite_real @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_867_sum__mono__inv,axiom,
! [F: risk_Free_account > real,I4: set_Ri1641125681238393385ccount,G: risk_Free_account > real,I: risk_Free_account] :
( ( ( groups8212752698507433307t_real @ F @ I4 )
= ( groups8212752698507433307t_real @ G @ I4 ) )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member5612106785598075018ccount @ I @ I4 )
=> ( ( finite1362240334998357386ccount @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_868_sum__mono__inv,axiom,
! [F: a > real,I4: set_a,G: a > real,I: a] :
( ( ( groups2740460157737275248a_real @ F @ I4 )
= ( groups2740460157737275248a_real @ G @ I4 ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_a @ I @ I4 )
=> ( ( finite_finite_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_869_sum__mono__inv,axiom,
! [F: int > real,I4: set_int,G: int > real,I: int] :
( ( ( groups8778361861064173332t_real @ F @ I4 )
= ( groups8778361861064173332t_real @ G @ I4 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I @ I4 )
=> ( ( finite_finite_int @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_870_sum__mono__inv,axiom,
! [F: real > risk_Free_account,I4: set_real,G: real > risk_Free_account,I: real] :
( ( ( groups8516999891779824987ccount @ F @ I4 )
= ( groups8516999891779824987ccount @ G @ I4 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I @ I4 )
=> ( ( finite_finite_real @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_871_sum__mono__inv,axiom,
! [F: risk_Free_account > risk_Free_account,I4: set_Ri1641125681238393385ccount,G: risk_Free_account > risk_Free_account,I: risk_Free_account] :
( ( ( groups5726723449141454370ccount @ F @ I4 )
= ( groups5726723449141454370ccount @ G @ I4 ) )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ I4 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member5612106785598075018ccount @ I @ I4 )
=> ( ( finite1362240334998357386ccount @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_872_sum__mono__inv,axiom,
! [F: nat > risk_Free_account,I4: set_nat,G: nat > risk_Free_account,I: nat] :
( ( ( groups6033208628184776703ccount @ F @ I4 )
= ( groups6033208628184776703ccount @ G @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_873_sum__mono__inv,axiom,
! [F: int > risk_Free_account,I4: set_int,G: int > risk_Free_account,I: int] :
( ( ( groups2220918773033463387ccount @ F @ I4 )
= ( groups2220918773033463387ccount @ G @ I4 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I4 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I @ I4 )
=> ( ( finite_finite_int @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_874_sum__mono__inv,axiom,
! [F: real > nat,I4: set_real,G: real > nat,I: real] :
( ( ( groups1935376822645274424al_nat @ F @ I4 )
= ( groups1935376822645274424al_nat @ G @ I4 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I @ I4 )
=> ( ( finite_finite_real @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_875_sum__mono__inv,axiom,
! [F: risk_Free_account > nat,I4: set_Ri1641125681238393385ccount,G: risk_Free_account > nat,I: risk_Free_account] :
( ( ( groups1716945928530811391nt_nat @ F @ I4 )
= ( groups1716945928530811391nt_nat @ G @ I4 ) )
=> ( ! [I3: risk_Free_account] :
( ( member5612106785598075018ccount @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member5612106785598075018ccount @ I @ I4 )
=> ( ( finite1362240334998357386ccount @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_876_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_877_sum_Ointer__filter,axiom,
! [A: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_878_sum_Ointer__filter,axiom,
! [A: set_Ri1641125681238393385ccount,G: risk_Free_account > real,P: risk_Free_account > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( groups8212752698507433307t_real @ G
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8212752698507433307t_real
@ ^ [X3: risk_Free_account] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_879_sum_Ointer__filter,axiom,
! [A: set_a,G: a > real,P: a > $o] :
( ( finite_finite_a @ A )
=> ( ( groups2740460157737275248a_real @ G
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups2740460157737275248a_real
@ ^ [X3: a] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_880_sum_Ointer__filter,axiom,
! [A: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_881_sum_Ointer__filter,axiom,
! [A: set_real,G: real > risk_Free_account,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8516999891779824987ccount @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8516999891779824987ccount
@ ^ [X3: real] : ( if_Risk_Free_account @ ( P @ X3 ) @ ( G @ X3 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_882_sum_Ointer__filter,axiom,
! [A: set_Ri1641125681238393385ccount,G: risk_Free_account > risk_Free_account,P: risk_Free_account > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( groups5726723449141454370ccount @ G
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups5726723449141454370ccount
@ ^ [X3: risk_Free_account] : ( if_Risk_Free_account @ ( P @ X3 ) @ ( G @ X3 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_883_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > risk_Free_account,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups6033208628184776703ccount
@ ^ [X3: nat] : ( if_Risk_Free_account @ ( P @ X3 ) @ ( G @ X3 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_884_sum_Ointer__filter,axiom,
! [A: set_int,G: int > risk_Free_account,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups2220918773033463387ccount @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups2220918773033463387ccount
@ ^ [X3: int] : ( if_Risk_Free_account @ ( P @ X3 ) @ ( G @ X3 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_885_sum_Ointer__filter,axiom,
! [A: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_886_sum_Ointer__filter,axiom,
! [A: set_Ri1641125681238393385ccount,G: risk_Free_account > nat,P: risk_Free_account > $o] :
( ( finite1362240334998357386ccount @ A )
=> ( ( groups1716945928530811391nt_nat @ G
@ ( collec1856553087948576712ccount
@ ^ [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups1716945928530811391nt_nat
@ ^ [X3: risk_Free_account] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_887_GreatestI__ex__nat,axiom,
! [P: nat > $o,B4: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_888_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_889_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_890_sum__le__included,axiom,
! [S2: set_int,T3: set_int,G: int > real,I: int > int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_891_sum__le__included,axiom,
! [S2: set_nat,T3: set_nat,G: nat > risk_Free_account,I: nat > nat,F: nat > risk_Free_account] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ S2 ) @ ( groups6033208628184776703ccount @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_892_sum__le__included,axiom,
! [S2: set_nat,T3: set_int,G: int > risk_Free_account,I: int > nat,F: nat > risk_Free_account] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ S2 ) @ ( groups2220918773033463387ccount @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_893_sum__le__included,axiom,
! [S2: set_int,T3: set_nat,G: nat > risk_Free_account,I: nat > int,F: int > risk_Free_account] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups2220918773033463387ccount @ F @ S2 ) @ ( groups6033208628184776703ccount @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_894_sum__le__included,axiom,
! [S2: set_int,T3: set_int,G: int > risk_Free_account,I: int > int,F: int > risk_Free_account] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups2220918773033463387ccount @ F @ S2 ) @ ( groups2220918773033463387ccount @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_895_sum__le__included,axiom,
! [S2: set_nat,T3: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_896_sum__le__included,axiom,
! [S2: set_nat,T3: set_int,G: int > nat,I: int > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_897_sum__le__included,axiom,
! [S2: set_int,T3: set_nat,G: nat > nat,I: nat > int,F: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_898_sum__le__included,axiom,
! [S2: set_int,T3: set_int,G: int > nat,I: int > int,F: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_899_sum__le__included,axiom,
! [S2: set_nat,T3: set_nat,G: nat > int,I: nat > nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_int @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S2 ) @ ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_900_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > real] :
( ( finite_finite_real @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A )
= zero_zero_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_901_sum__nonneg__eq__0__iff,axiom,
! [A: set_Ri1641125681238393385ccount,F: risk_Free_account > real] :
( ( finite1362240334998357386ccount @ A )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups8212752698507433307t_real @ F @ A )
= zero_zero_real )
= ( ! [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_902_sum__nonneg__eq__0__iff,axiom,
! [A: set_a,F: a > real] :
( ( finite_finite_a @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups2740460157737275248a_real @ F @ A )
= zero_zero_real )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_903_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > real] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A )
= zero_zero_real )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_904_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > risk_Free_account] :
( ( finite_finite_real @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ( ( groups8516999891779824987ccount @ F @ A )
= zero_z1425366712893667068ccount )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( F @ X3 )
= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_905_sum__nonneg__eq__0__iff,axiom,
! [A: set_Ri1641125681238393385ccount,F: risk_Free_account > risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ( ( groups5726723449141454370ccount @ F @ A )
= zero_z1425366712893667068ccount )
= ( ! [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
=> ( ( F @ X3 )
= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_906_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > risk_Free_account] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ( ( groups6033208628184776703ccount @ F @ A )
= zero_z1425366712893667068ccount )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_907_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > risk_Free_account] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ( ( groups2220918773033463387ccount @ F @ A )
= zero_z1425366712893667068ccount )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( F @ X3 )
= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_908_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > nat] :
( ( finite_finite_real @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A )
= zero_zero_nat )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_909_sum__nonneg__eq__0__iff,axiom,
! [A: set_Ri1641125681238393385ccount,F: risk_Free_account > nat] :
( ( finite1362240334998357386ccount @ A )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups1716945928530811391nt_nat @ F @ A )
= zero_zero_nat )
= ( ! [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_910_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_911_of__nat__sum,axiom,
! [F: real > nat,A: set_real] :
( ( semiri1314217659103216013at_int @ ( groups1935376822645274424al_nat @ F @ A ) )
= ( groups1932886352136224148al_int
@ ^ [X3: real] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_912_of__nat__sum,axiom,
! [F: ( nat > real ) > nat,A: set_nat_real] :
( ( semiri5074537144036343181t_real @ ( groups780007972294800423al_nat @ F @ A ) )
= ( groups4253619806861319043l_real
@ ^ [X3: nat > real] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_913_of__nat__sum,axiom,
! [F: ( nat > real ) > nat,A: set_nat_real] :
( ( semiri1316708129612266289at_nat @ ( groups780007972294800423al_nat @ F @ A ) )
= ( groups780007972294800423al_nat
@ ^ [X3: nat > real] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_914_of__nat__sum,axiom,
! [F: ( nat > real ) > nat,A: set_nat_real] :
( ( semiri1314217659103216013at_int @ ( groups780007972294800423al_nat @ F @ A ) )
= ( groups777517501785750147al_int
@ ^ [X3: nat > real] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_915_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_916_finite__Collect__subsets,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B2: set_int] : ( ord_less_eq_set_int @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_917_finite__Collect__subsets,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_918_finite__Collect__subsets,axiom,
! [A: set_Ri1641125681238393385ccount] :
( ( finite1362240334998357386ccount @ A )
=> ( finite8674737585558172096ccount
@ ( collec2171358002926308350ccount
@ ^ [B2: set_Ri1641125681238393385ccount] : ( ord_le4487465848215339657ccount @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_919_finite__Collect__subsets,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( finite9007344921179782393t_real
@ ( collect_set_real
@ ^ [B2: set_real] : ( ord_less_eq_set_real @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_920_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
~ ( P @ X3 ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_921_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_922_finite__Collect__not,axiom,
! [P: int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
~ ( P @ X3 ) ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Collect_not
thf(fact_923_finite__Collect__not,axiom,
! [P: product_unit > $o] :
( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
~ ( P @ X3 ) ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Collect_not
thf(fact_924_finite__Collect__not,axiom,
! [P: set_Product_unit > $o] :
( ( finite1772178364199683094t_unit @ ( collec7787489603430924120t_unit @ P ) )
=> ( ( finite1772178364199683094t_unit
@ ( collec7787489603430924120t_unit
@ ^ [X3: set_Product_unit] :
~ ( P @ X3 ) ) )
= ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit ) ) ) ).
% finite_Collect_not
thf(fact_925_finite__Collect__not,axiom,
! [P: set_int > $o] :
( ( finite6197958912794628473et_int @ ( collect_set_int @ P ) )
=> ( ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [X3: set_int] :
~ ( P @ X3 ) ) )
= ( finite6197958912794628473et_int @ top_top_set_set_int ) ) ) ).
% finite_Collect_not
thf(fact_926_finite__Collect__not,axiom,
! [P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
~ ( P @ X3 ) ) )
= ( finite1152437895449049373et_nat @ top_top_set_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_927_finite__Collect__not,axiom,
! [P: set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
~ ( P @ X3 ) ) )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ) ).
% finite_Collect_not
thf(fact_928_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_929_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_930_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_931_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_932_finite__option__UNIV,axiom,
( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% finite_option_UNIV
thf(fact_933_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_934_finite__option__UNIV,axiom,
( ( finite4751406903043366492t_unit @ top_to6468764214823621219t_unit )
= ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit ) ) ).
% finite_option_UNIV
thf(fact_935_finite__option__UNIV,axiom,
( ( finite4104298959215839049et_int @ top_to8332679073461343512et_int )
= ( finite6197958912794628473et_int @ top_top_set_set_int ) ) ).
% finite_option_UNIV
thf(fact_936_finite__option__UNIV,axiom,
( ( finite6594382104147565805et_nat @ top_to3922855455394486460et_nat )
= ( finite1152437895449049373et_nat @ top_top_set_set_nat ) ) ).
% finite_option_UNIV
thf(fact_937_finite__option__UNIV,axiom,
( ( finite3831083272032232269_set_a @ top_to3949272007228979924_set_a )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ).
% finite_option_UNIV
thf(fact_938_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_939_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_940_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_941_finite__Plus__UNIV__iff,axiom,
( ( finite2069262655233506379t_unit @ top_to1755696212014396186t_unit )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_942_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_943_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_944_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_945_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_946_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_947_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_948_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_949_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_950_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_951_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
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_952_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
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_953_abs__abs,axiom,
! [A4: real] :
( ( abs_abs_real @ ( abs_abs_real @ A4 ) )
= ( abs_abs_real @ A4 ) ) ).
% abs_abs
thf(fact_954_abs__abs,axiom,
! [A4: int] :
( ( abs_abs_int @ ( abs_abs_int @ A4 ) )
= ( abs_abs_int @ A4 ) ) ).
% abs_abs
thf(fact_955_finite__atLeastAtMost,axiom,
! [L2: nat,U2: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) ) ).
% finite_atLeastAtMost
thf(fact_956_int__sum,axiom,
! [F: real > nat,A: set_real] :
( ( semiri1314217659103216013at_int @ ( groups1935376822645274424al_nat @ F @ A ) )
= ( groups1932886352136224148al_int
@ ^ [X3: real] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_957_int__sum,axiom,
! [F: ( nat > real ) > nat,A: set_nat_real] :
( ( semiri1314217659103216013at_int @ ( groups780007972294800423al_nat @ F @ A ) )
= ( groups777517501785750147al_int
@ ^ [X3: nat > real] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_958_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_959_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_960_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_12: nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_961_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_12: int] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_962_pigeonhole__infinite__rel,axiom,
! [A: set_real,B: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_963_pigeonhole__infinite__rel,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_nat,R: risk_Free_account > nat > $o] :
( ~ ( finite1362240334998357386ccount @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite1362240334998357386ccount
@ ( collec1856553087948576712ccount
@ ^ [A5: risk_Free_account] :
( ( member5612106785598075018ccount @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_964_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_965_pigeonhole__infinite__rel,axiom,
! [A: set_real,B: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_966_pigeonhole__infinite__rel,axiom,
! [A: set_Ri1641125681238393385ccount,B: set_int,R: risk_Free_account > int > $o] :
( ~ ( finite1362240334998357386ccount @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite1362240334998357386ccount
@ ( collec1856553087948576712ccount
@ ^ [A5: risk_Free_account] :
( ( member5612106785598075018ccount @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_967_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_int,R: a > int > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_968_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_969_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_970_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_971_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_972_finite__has__maximal2,axiom,
! [A: set_real,A4: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A4 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ A4 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_973_finite__has__maximal2,axiom,
! [A: set_Ri1641125681238393385ccount,A4: risk_Free_account] :
( ( finite1362240334998357386ccount @ A )
=> ( ( member5612106785598075018ccount @ A4 @ A )
=> ? [X2: risk_Free_account] :
( ( member5612106785598075018ccount @ X2 @ A )
& ( ord_le4245800335709223507ccount @ A4 @ X2 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A )
=> ( ( ord_le4245800335709223507ccount @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_974_finite__has__maximal2,axiom,
! [A: set_nat,A4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A4 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A4 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_975_finite__has__maximal2,axiom,
! [A: set_set_nat,A4: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A4 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A4 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_976_finite__has__maximal2,axiom,
! [A: set_int,A4: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A4 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A4 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_977_finite__has__maximal2,axiom,
! [A: set_nat_real,A4: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A4 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ A4 @ X2 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_978_finite__has__maximal2,axiom,
! [A: set_set_int,A4: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( member_set_int @ A4 @ A )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( ord_less_eq_set_int @ A4 @ X2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A )
=> ( ( ord_less_eq_set_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_979_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A4: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A4 @ A )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ( ord_le6893508408891458716et_nat @ A4 @ X2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_980_finite__has__maximal2,axiom,
! [A: set_se7412911595254129503ccount,A4: set_Ri1641125681238393385ccount] :
( ( finite8674737585558172096ccount @ A )
=> ( ( member8751979616678796480ccount @ A4 @ A )
=> ? [X2: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ X2 @ A )
& ( ord_le4487465848215339657ccount @ A4 @ X2 )
& ! [Xa: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ Xa @ A )
=> ( ( ord_le4487465848215339657ccount @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_981_finite__has__maximal2,axiom,
! [A: set_set_real,A4: set_real] :
( ( finite9007344921179782393t_real @ A )
=> ( ( member_set_real @ A4 @ A )
=> ? [X2: set_real] :
( ( member_set_real @ X2 @ A )
& ( ord_less_eq_set_real @ A4 @ X2 )
& ! [Xa: set_real] :
( ( member_set_real @ Xa @ A )
=> ( ( ord_less_eq_set_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_982_finite__has__minimal2,axiom,
! [A: set_nat_real,A4: nat > real] :
( ( finite7853608736407863218t_real @ A )
=> ( ( member_nat_real @ A4 @ A )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A )
& ( ord_less_eq_nat_real @ X2 @ A4 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A )
=> ( ( ord_less_eq_nat_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_983_finite__has__minimal2,axiom,
! [A: set_set_int,A4: set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( member_set_int @ A4 @ A )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( ord_less_eq_set_int @ X2 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A )
=> ( ( ord_less_eq_set_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_984_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A4: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A4 @ A )
=> ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ( ord_le6893508408891458716et_nat @ X2 @ A4 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_985_finite__has__minimal2,axiom,
! [A: set_se7412911595254129503ccount,A4: set_Ri1641125681238393385ccount] :
( ( finite8674737585558172096ccount @ A )
=> ( ( member8751979616678796480ccount @ A4 @ A )
=> ? [X2: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ X2 @ A )
& ( ord_le4487465848215339657ccount @ X2 @ A4 )
& ! [Xa: set_Ri1641125681238393385ccount] :
( ( member8751979616678796480ccount @ Xa @ A )
=> ( ( ord_le4487465848215339657ccount @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_986_finite__has__minimal2,axiom,
! [A: set_set_real,A4: set_real] :
( ( finite9007344921179782393t_real @ A )
=> ( ( member_set_real @ A4 @ A )
=> ? [X2: set_real] :
( ( member_set_real @ X2 @ A )
& ( ord_less_eq_set_real @ X2 @ A4 )
& ! [Xa: set_real] :
( ( member_set_real @ Xa @ A )
=> ( ( ord_less_eq_set_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_987_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_988_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_989_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_990_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_991_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_992_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_993_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_994_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_995_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_996_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_997_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_998_finite__interval__int1,axiom,
! [A4: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A4 @ I2 )
& ( ord_less_eq_int @ I2 @ B4 ) ) ) ) ).
% finite_interval_int1
thf(fact_999_finite__interval__int3,axiom,
! [A4: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A4 @ I2 )
& ( ord_less_eq_int @ I2 @ B4 ) ) ) ) ).
% finite_interval_int3
thf(fact_1000_finite__interval__int2,axiom,
! [A4: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A4 @ I2 )
& ( ord_less_int @ I2 @ B4 ) ) ) ) ).
% finite_interval_int2
thf(fact_1001_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_1002_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1003_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1004_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1005_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1006_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1007_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1008_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1009_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1010_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1011_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1012_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1013_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1014_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1015_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( K
!= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% nonneg_int_cases
thf(fact_1016_verit__la__generic,axiom,
! [A4: int,X: int] :
( ( ord_less_eq_int @ A4 @ X )
| ( A4 = X )
| ( ord_less_eq_int @ X @ A4 ) ) ).
% verit_la_generic
thf(fact_1017_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1018_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1019_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1020_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N5 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1021_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_1022_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1023_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_1024_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1025_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1026_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_1027_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1028_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_1029_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1030_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1031_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1032_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1033_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_1034_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1035_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1036_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1037_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_1038_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1039_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1040_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1041_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_1042_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_1043_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_1044_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1045_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N5: nat] :
( ~ ( P @ N5 )
& ( P @ ( suc @ N5 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1046_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_1047_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1048_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1049_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1050_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1051_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1052_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1053_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1054_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1055_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1056_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1057_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1058_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1059_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1060_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z4: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( ! [N5: nat] : ( R @ N5 @ ( suc @ N5 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1061_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1062_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N5 )
=> ( P @ M6 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1063_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1064_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1065_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1066_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1067_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1068_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1069_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1070_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1071_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1072_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1073_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1074_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1075_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1076_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1077_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_1078_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1079_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1080_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1081_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1082_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1083_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1084_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1085_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1086_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_1087_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1088_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1089_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1090_finite__atLeastAtMost__int,axiom,
! [L2: int,U2: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U2 ) ) ).
% finite_atLeastAtMost_int
thf(fact_1091_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1092_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1093_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1094_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1095_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_1096_finite__interval__int4,axiom,
! [A4: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A4 @ I2 )
& ( ord_less_int @ I2 @ B4 ) ) ) ) ).
% finite_interval_int4
thf(fact_1097_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1098_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1099_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1100_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_1101_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_1102_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_1103_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1104_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1105_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1106_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_1107_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1108_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1109_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_1110_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1111_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1112_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
=> ( P @ M6 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1113_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1114_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_1115_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_1116_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1117_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1118_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1119_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1120_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1121_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1122_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1123_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1124_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1125_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1126_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1127_le__diff__iff_H,axiom,
! [A4: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A4 ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1128_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_1129_diff__less__mono,axiom,
! [A4: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1130_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1131_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N5: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% int_diff_cases
thf(fact_1132_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_1133_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1134_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z5: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
& ! [Z5: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1135_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1136_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P2: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1137_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P2: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1138_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K4 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K4 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1139_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1140_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1141_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_1142_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1143_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1144_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1145_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_1146_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1147_Bolzano,axiom,
! [A4: real,B4: real,P: real > real > $o] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ! [A3: real,B3: real,C4: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C4 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C4 )
=> ( P @ A3 @ C4 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A4 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B4 )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X2 )
& ( ord_less_eq_real @ X2 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D2 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% Bolzano
thf(fact_1148_lemma__interval,axiom,
! [A4: real,X: real,B4: real] :
( ( ord_less_real @ A4 @ X )
=> ( ( ord_less_real @ X @ B4 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_eq_real @ A4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ B4 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1149_lemma__interval__lt,axiom,
! [A4: real,X: real,B4: real] :
( ( ord_less_real @ A4 @ X )
=> ( ( ord_less_real @ X @ B4 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_real @ A4 @ Y5 )
& ( ord_less_real @ Y5 @ B4 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1150_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1151_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1152_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_1153_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_1154_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1155_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ 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 @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1156_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_1157_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1158_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1159_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1160_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1161_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1162_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1163_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1164_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1165_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1166_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1167_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1168_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1169_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1170_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1171_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1172_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1173_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1174_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1175_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_1176_card__atMost,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U2 ) )
= ( suc @ U2 ) ) ).
% card_atMost
thf(fact_1177_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1178_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1179_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1180_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1181_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_1182_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_1183_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_1184_card__atLeastAtMost,axiom,
! [L2: nat,U2: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) )
= ( minus_minus_nat @ ( suc @ U2 ) @ L2 ) ) ).
% card_atLeastAtMost
thf(fact_1185_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_1186_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_1187_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1188_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1189_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1190_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1191_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1192_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1193_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1194_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1195_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1196_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1197_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1198_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1199_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_1200_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1201_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times_int @ zero_zero_int @ L2 )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1202_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1203_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1204_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1205_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1206_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A4: nat] :
( ( A
= ( plus_plus_nat @ K @ A4 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A4 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1207_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1208_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1209_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1210_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1211_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1212_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1213_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1214_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N5: nat] :
( L2
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1215_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1216_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_1217_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1218_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1219_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1220_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1221_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1222_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1223_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1224_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1225_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1226_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1227_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1228_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1229_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1230_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1231_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1232_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1233_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1234_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1235_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1236_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1237_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1238_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_1239_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1240_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1241_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1242_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1243_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1244_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N5: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N5 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1245_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1246_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1247_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1248_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1249_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1250_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1251_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1259_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1260_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1261_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1262_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1263_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_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,
! [N5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ ( groups4655409347963886775ccount @ l @ top_top_set_a ) ) @ ( set_ord_atMost_nat @ N5 ) ) ) ).
%------------------------------------------------------------------------------