TPTP Problem File: SLH0557^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Risk_Free_Lending/0000_Risk_Free_Lending/prob_00193_006396__5748370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1389 ( 454 unt; 114 typ; 0 def)
% Number of atoms : 4870 (1519 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 15060 ( 383 ~; 80 |; 322 &;11686 @)
% ( 0 <=>;2589 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 1067 (1067 >; 0 *; 0 +; 0 <<)
% Number of symbols : 110 ( 107 usr; 11 con; 0-3 aty)
% Number of variables : 4470 ( 477 ^;3860 !; 133 ?;4470 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:56:32.591
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (107)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
invers8013647133539491842omplex: complex > complex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Risk____Free____Lending__Oaccount,type,
minus_4846202936726426316ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Risk____Free____Lending__Oaccount,type,
plus_p1863581527469039996ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Risk____Free____Lending__Oaccount,type,
uminus3377898441596595772ccount: risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Risk____Free____Lending__Oaccount,type,
zero_z1425366712893667068ccount: risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Risk____Free____Lending__Oaccount,type,
groups6033208628184776703ccount: ( nat > risk_Free_account ) > set_nat > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Risk____Free____Lending__Oaccount,type,
groups8516999891779824987ccount: ( real > risk_Free_account ) > set_real > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Complex__Ocomplex,type,
groups8255218700646806128omplex: ( set_nat > complex ) > set_set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
groups5107569545109728110t_real: ( set_nat > real ) > set_set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Risk____Free____Lending__Oaccount,type,
groups5807469391267537845ccount: ( set_nat > risk_Free_account ) > set_set_nat > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Complex__Ocomplex,type,
groups1092910753850256091omplex: ( set_nat > complex ) > set_set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4248547760180025341at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
groups3619160379726066777t_real: ( set_nat > real ) > set_set_nat > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001t__Risk____Free____Lending__Oaccount,type,
if_Risk_Free_account: $o > risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Real__Vector__Spaces_Odependent_001t__Real__Oreal,type,
real_V7051607973971999986t_real: set_real > $o ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
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__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
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_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_x____,type,
x: risk_Free_account ).
thf(sy_v_y____,type,
y: risk_Free_account ).
% Relevant facts (1267)
thf(fact_0__092_060open_062x_A_092_060le_062_Ay_092_060close_062,axiom,
ord_le4245800335709223507ccount @ x @ y ).
% \<open>x \<le> y\<close>
thf(fact_1__092_060open_062y_A_092_060le_062_Ax_092_060close_062,axiom,
ord_le4245800335709223507ccount @ y @ x ).
% \<open>y \<le> x\<close>
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_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_4_atMost__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( set_ord_atMost_real @ X )
= ( set_ord_atMost_real @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_5_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_6_sum_Oswap,axiom,
! [G: nat > real > real,B: set_real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups8097168146408367636l_real @ ( G @ I ) @ B )
@ A )
= ( groups8097168146408367636l_real
@ ^ [J: real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_7_sum_Oswap,axiom,
! [G: real > nat > real,B: set_nat,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups8097168146408367636l_real
@ ^ [I: real] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_8_sum_Oswap,axiom,
! [G: real > real > real,B: set_real,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : ( groups8097168146408367636l_real @ ( G @ I ) @ B )
@ A )
= ( groups8097168146408367636l_real
@ ^ [J: real] :
( groups8097168146408367636l_real
@ ^ [I: real] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_9_sum_Oswap,axiom,
! [G: nat > nat > real,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_10_sum_Ocong,axiom,
! [A: set_real,B: set_real,G: real > real,H: real > real] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ A )
= ( groups8097168146408367636l_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_11_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_12_sum_Oeq__general,axiom,
! [B: set_nat,A: set_set_nat,H: set_nat > nat,Gamma: nat > real,Phi: set_nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: set_nat] :
( ( ( member_set_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5107569545109728110t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_13_sum_Oeq__general,axiom,
! [B: set_real,A: set_set_nat,H: set_nat > real,Gamma: real > real,Phi: set_nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: set_nat] :
( ( ( member_set_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5107569545109728110t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_14_sum_Oeq__general,axiom,
! [B: set_set_nat,A: set_nat,H: nat > set_nat,Gamma: set_nat > real,Phi: nat > real] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups5107569545109728110t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_15_sum_Oeq__general,axiom,
! [B: set_set_nat,A: set_real,H: real > set_nat,Gamma: set_nat > real,Phi: real > real] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups5107569545109728110t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_16_sum_Oeq__general,axiom,
! [B: set_real,A: set_real,H: real > real,Gamma: real > real,Phi: real > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_17_sum_Oeq__general,axiom,
! [B: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [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_18_sum_Oeq__general,axiom,
! [B: set_real,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_19_sum_Oeq__general,axiom,
! [B: set_nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_20_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > set_nat,A: set_set_nat,H: set_nat > nat,Gamma: nat > real,Phi: set_nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_set_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5107569545109728110t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_21_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > set_nat,A: set_set_nat,H: set_nat > real,Gamma: real > real,Phi: set_nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ( ( member_set_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5107569545109728110t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_22_sum_Oeq__general__inverses,axiom,
! [B: set_set_nat,K: set_nat > nat,A: set_nat,H: nat > set_nat,Gamma: set_nat > real,Phi: nat > real] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups5107569545109728110t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_23_sum_Oeq__general__inverses,axiom,
! [B: set_set_nat,K: set_nat > real,A: set_real,H: real > set_nat,Gamma: set_nat > real,Phi: real > real] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups5107569545109728110t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_24_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > real,A: set_real,H: real > real,Gamma: real > real,Phi: real > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_25_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_real @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_26_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_27_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_28_sum_Oreindex__bij__witness,axiom,
! [S: set_set_nat,I2: nat > set_nat,J2: set_nat > nat,T: set_nat,H: nat > real,G: set_nat > real] :
( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_set_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5107569545109728110t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_29_sum_Oreindex__bij__witness,axiom,
! [S: set_set_nat,I2: real > set_nat,J2: set_nat > real,T: set_real,H: real > real,G: set_nat > real] :
( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( member_real @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( member_set_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5107569545109728110t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_30_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: set_nat > nat,J2: nat > set_nat,T: set_set_nat,H: set_nat > real,G: nat > real] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_set_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups5107569545109728110t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_31_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: set_nat > real,J2: real > set_nat,T: set_set_nat,H: set_nat > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_set_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( member_real @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups5107569545109728110t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_32_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: real > real,J2: real > real,T: set_real,H: real > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_real @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( member_real @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_33_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J2: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_real @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_34_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J2: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_real @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_35_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_36_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_37_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_38_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_39_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_40_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_41_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_42_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_43_sum__mono,axiom,
! [K2: set_set_nat,F: set_nat > real,G: set_nat > real] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups5107569545109728110t_real @ F @ K2 ) @ ( groups5107569545109728110t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_44_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_45_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_46_sum__mono,axiom,
! [K2: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ K2 ) @ ( groups8294997508430121362at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_47_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_48_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_49_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% less_eq_account_def
thf(fact_50_of__real__sum,axiom,
! [F: nat > real,S2: set_nat] :
( ( real_V1803761363581548252l_real @ ( groups6591440286371151544t_real @ F @ S2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1803761363581548252l_real @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_51_of__real__sum,axiom,
! [F: real > real,S2: set_real] :
( ( real_V1803761363581548252l_real @ ( groups8097168146408367636l_real @ F @ S2 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1803761363581548252l_real @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_52_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_53_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_54_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_55_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_56_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_57_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_58_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_60_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_61_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_62_norm__sum,axiom,
! [F: nat > complex,A: set_nat] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A ) ) ).
% norm_sum
thf(fact_63_norm__sum,axiom,
! [F: real > complex,A: set_real] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ A ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A ) ) ).
% norm_sum
thf(fact_64_norm__sum,axiom,
! [F: nat > real,A: set_nat] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V7735802525324610683m_real @ ( F @ I ) )
@ A ) ) ).
% norm_sum
thf(fact_65_norm__sum,axiom,
! [F: real > real,A: set_real] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups8097168146408367636l_real @ F @ A ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V7735802525324610683m_real @ ( F @ I ) )
@ A ) ) ).
% norm_sum
thf(fact_66_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ? [Y2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y2 ) )
& ! [Z: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_67_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_68_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_69_verit__comp__simplify1_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_70_nle__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_71_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_72_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_73_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_74_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_75_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_76_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_77_ord__eq__le__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( A3 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_78_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_79_ord__eq__le__trans,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_80_ord__le__eq__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_81_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_82_ord__le__eq__trans,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_83_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_84_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_85_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_86_dual__order_Oantisym,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_87_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_88_dual__order_Oantisym,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_89_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_90_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [Uu: real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_91_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_92_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_93_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_94_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_95_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_96_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_97_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_98_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_99_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_100_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_101_sum_Oneutral,axiom,
! [A: set_real,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_102_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > real,A: set_set_nat] :
( ( ( groups5107569545109728110t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_103_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_104_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > risk_Free_account,A: set_nat] :
( ( ( groups6033208628184776703ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_105_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > risk_Free_account,A: set_set_nat] :
( ( ( groups5807469391267537845ccount @ G @ A )
!= zero_z1425366712893667068ccount )
=> ~ ! [A2: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_106_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_107_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_108_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > nat,A: set_set_nat] :
( ( ( groups8294997508430121362at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_109_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_110_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_111_sum__nonneg,axiom,
! [A: set_set_nat,F: set_nat > real] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5107569545109728110t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_112_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_113_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_114_sum__nonneg,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups8294997508430121362at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_115_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_116_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_117_sum__nonpos,axiom,
! [A: set_set_nat,F: set_nat > real] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5107569545109728110t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_118_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_119_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_120_sum__nonpos,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_121_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_122_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_123_sum__norm__le,axiom,
! [S: set_set_nat,F: set_nat > complex,G: set_nat > real] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_124_sum__norm__le,axiom,
! [S: set_nat,F: nat > complex,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_125_sum__norm__le,axiom,
! [S: set_real,F: real > complex,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_126_sum__norm__le,axiom,
! [S: set_nat,F: nat > real,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_127_sum__norm__le,axiom,
! [S: set_real,F: real > real,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups8097168146408367636l_real @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_128_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_129_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_130_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_131_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_132_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_133_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_134_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_137_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_138_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_139_ord__le__eq__subst,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_140_ord__le__eq__subst,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_143_mem__Collect__eq,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_145_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_146_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_148_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_149_ord__eq__le__subst,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A3: set_nat,F: real > set_nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_ord__eq__le__subst,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_155_ord__eq__le__subst,axiom,
! [A3: real,F: set_nat > real,B3: set_nat,C: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_156_ord__eq__le__subst,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_157_ord__eq__le__subst,axiom,
! [A3: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_158_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_159_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_160_verit__la__disequality,axiom,
! [A3: real,B3: real] :
( ( A3 = B3 )
| ~ ( ord_less_eq_real @ A3 @ B3 )
| ~ ( ord_less_eq_real @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_161_verit__la__disequality,axiom,
! [A3: nat,B3: nat] :
( ( A3 = B3 )
| ~ ( ord_less_eq_nat @ A3 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_162_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_163_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_164_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_165_order__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_166_order__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_167_order__subst2,axiom,
! [A3: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_168_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_172_order__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_173_order__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_174_order__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_order__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_176_order__subst1,axiom,
! [A3: real,F: set_nat > real,B3: set_nat,C: set_nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A3: set_nat,F: real > set_nat,B3: real,C: real] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst1,axiom,
! [A3: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_184_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_185_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_186_antisym,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_187_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_188_antisym,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_189_dual__order_Otrans,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_191_dual__order_Otrans,axiom,
! [B3: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_192_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_193_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_194_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_195_linorder__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_196_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_197_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_198_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_199_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_200_order_Otrans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% order.trans
thf(fact_201_order_Otrans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_202_order_Otrans,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_203_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_204_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_205_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_206_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_207_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_208_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_209_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_210_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_211_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_212_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_213_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_214_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_215_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_216_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_217_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ R )
@ ^ [X4: real] : ( member_real @ X4 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_218_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_219_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R )
@ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_220_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_221_prop__restrict,axiom,
! [X: real,Z4: set_real,X5: set_real,P: real > $o] :
( ( member_real @ X @ Z4 )
=> ( ( ord_less_eq_set_real @ Z4
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_222_prop__restrict,axiom,
! [X: set_nat,Z4: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ Z4 )
=> ( ( ord_le6893508408891458716et_nat @ Z4
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_223_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_224_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_225_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_226_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_227_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_228_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_229_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_230_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_231_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_232_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_233_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_234_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_235_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A5 )
=> ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_236_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_237_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_238_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_239_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A5 )
=> ( member_real @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_240_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A5 )
=> ( member_set_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_241_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_242_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_243_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_244_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_245_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_246_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_247_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_248_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_249_Collect__restrict,axiom,
! [X5: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_250_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_251_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_252_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
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: real] :
( ( member_real @ X4 @ B )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_253_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
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_254_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
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_255_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
@ ^ [X4: real] :
( ( member_real @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_256_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
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_257_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
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_258_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_259_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_260_scaleR__mono,axiom,
! [A3: real,B3: real,X: real,Y: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ B3 @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_261_scaleR__mono_H,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ C ) @ ( real_V1485227260804924795R_real @ B3 @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_262_split__scaleR__neg__le,axiom,
! [A3: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_263_split__scaleR__pos__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A3 @ B3 ) ) ) ).
% split_scaleR_pos_le
thf(fact_264_scaleR__nonneg__nonneg,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A3 @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_265_scaleR__nonneg__nonpos,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_266_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_267_scaleR__zero__right,axiom,
! [A3: real] :
( ( real_V1485227260804924795R_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_268_scaleR__cancel__right,axiom,
! [A3: real,X: real,B3: real] :
( ( ( real_V1485227260804924795R_real @ A3 @ X )
= ( real_V1485227260804924795R_real @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_269_scaleR__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_270_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_271_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_272_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_273_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_274_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_275_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_276_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y ) ) ) ).
% dense
thf(fact_277_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_278_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_279_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_280_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_281_ord__eq__less__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( A3 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_282_ord__eq__less__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_283_ord__less__eq__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_284_ord__less__eq__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_285_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_286_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_287_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_288_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_289_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_290_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_291_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_292_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_293_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_294_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_295_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_296_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_297_order_Ostrict__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_298_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_299_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_300_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_301_dual__order_Ostrict__trans,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_302_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_303_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_304_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_305_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_306_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_307_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_308_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_309_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_310_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_311_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_312_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_313_order__less__asym_H,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_314_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_315_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_316_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_317_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_318_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_319_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_320_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_321_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_322_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_323_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_324_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_325_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_326_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_327_order__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_328_order__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_329_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_330_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_331_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_332_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_333_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_334_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_335_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_336_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_337_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_338_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_339_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_340_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_341_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_342_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_343_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_344_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_345_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_346_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_347_scaleR__le__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( real_V1485227260804924795R_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_348_scaleR__le__cancel__left__neg,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( real_V1485227260804924795R_real @ C @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_349_scaleR__le__cancel__left__pos,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( real_V1485227260804924795R_real @ C @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_350_scaleR__right__imp__eq,axiom,
! [X: real,A3: real,B3: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A3 @ X )
= ( real_V1485227260804924795R_real @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_351_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_352_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_353_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_354_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_355_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_356_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_357_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_358_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_359_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_360_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_361_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_362_order__less__le__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_363_order__less__le__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_364_order__less__le__subst1,axiom,
! [A3: set_nat,F: real > set_nat,B3: real,C: real] :
( ( ord_less_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_365_order__less__le__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_366_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_367_order__less__le__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_368_order__less__le__subst1,axiom,
! [A3: real,F: set_nat > real,B3: set_nat,C: set_nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_369_order__less__le__subst1,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_370_order__less__le__subst1,axiom,
! [A3: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_371_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_372_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_373_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_374_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_375_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_376_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_377_order__le__less__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > real,C: real] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_378_order__le__less__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_379_order__le__less__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_380_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_381_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_382_order__le__less__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_383_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_384_order__le__less__subst1,axiom,
! [A3: set_nat,F: real > set_nat,B3: real,C: real] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_385_order__le__less__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_386_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_387_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_388_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_389_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_390_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_391_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_392_order__neq__le__trans,axiom,
! [A3: real,B3: real] :
( ( A3 != B3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_393_order__neq__le__trans,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_394_order__neq__le__trans,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_395_order__le__neq__trans,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_396_order__le__neq__trans,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_397_order__le__neq__trans,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_398_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_399_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_400_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_401_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_402_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_403_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_404_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_405_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_406_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_407_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_408_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_409_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_410_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_411_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_412_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_413_dual__order_Ostrict__implies__order,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ( ord_less_eq_set_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_414_order_Ostrict__implies__order,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_415_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_416_order_Ostrict__implies__order,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_417_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_418_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_419_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_420_dual__order_Ostrict__trans2,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_421_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_422_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_423_dual__order_Ostrict__trans1,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_424_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_425_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_426_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_427_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_428_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_429_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_430_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_431_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_432_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_433_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_434_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_435_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_436_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_437_order_Ostrict__trans2,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_438_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_439_order_Ostrict__trans2,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_440_order_Ostrict__trans1,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_441_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_442_order_Ostrict__trans1,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_443_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_444_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_445_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_446_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_447_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_448_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_449_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_450_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_451_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_452_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_453_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_454_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_455_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_456_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_457_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_458_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_459_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_460_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_461_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_462_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_463_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_464_nless__le,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ~ ( ord_less_set_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_465_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_466_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_467_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_468_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_469_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_470_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_471_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_472_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_473_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_474_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_475_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_476_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_477_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_478_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_479_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_480_scaleR__le__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( A3 = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_481_zero__le__scaleR__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( A3 = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_482_scaleR__right_Osum,axiom,
! [A3: real,G: nat > real,A: set_nat] :
( ( real_V1485227260804924795R_real @ A3 @ ( groups6591440286371151544t_real @ G @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ A3 @ ( G @ X4 ) )
@ A ) ) ).
% scaleR_right.sum
thf(fact_483_scaleR__right_Osum,axiom,
! [A3: real,G: real > real,A: set_real] :
( ( real_V1485227260804924795R_real @ A3 @ ( groups8097168146408367636l_real @ G @ A ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ A3 @ ( G @ X4 ) )
@ A ) ) ).
% scaleR_right.sum
thf(fact_484_scaleR__sum__right,axiom,
! [A3: real,F: nat > real,A: set_nat] :
( ( real_V1485227260804924795R_real @ A3 @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ A3 @ ( F @ X4 ) )
@ A ) ) ).
% scaleR_sum_right
thf(fact_485_scaleR__sum__right,axiom,
! [A3: real,F: real > real,A: set_real] :
( ( real_V1485227260804924795R_real @ A3 @ ( groups8097168146408367636l_real @ F @ A ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ A3 @ ( F @ X4 ) )
@ A ) ) ).
% scaleR_sum_right
thf(fact_486_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_487_scaleR__left_Osum,axiom,
! [G: nat > real,A: set_nat,X: real] :
( ( real_V1485227260804924795R_real @ ( groups6591440286371151544t_real @ G @ A ) @ X )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ ( G @ X4 ) @ X )
@ A ) ) ).
% scaleR_left.sum
thf(fact_488_scaleR__left_Osum,axiom,
! [G: real > real,A: set_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups8097168146408367636l_real @ G @ A ) @ X )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( G @ X4 ) @ X )
@ A ) ) ).
% scaleR_left.sum
thf(fact_489_scaleR__sum__left,axiom,
! [F: nat > real,A: set_nat,X: real] :
( ( real_V1485227260804924795R_real @ ( groups6591440286371151544t_real @ F @ A ) @ X )
= ( groups6591440286371151544t_real
@ ^ [A4: nat] : ( real_V1485227260804924795R_real @ ( F @ A4 ) @ X )
@ A ) ) ).
% scaleR_sum_left
thf(fact_490_scaleR__sum__left,axiom,
! [F: real > real,A: set_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups8097168146408367636l_real @ F @ A ) @ X )
= ( groups8097168146408367636l_real
@ ^ [A4: real] : ( real_V1485227260804924795R_real @ ( F @ A4 ) @ X )
@ A ) ) ).
% scaleR_sum_left
thf(fact_491_scaleR__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ C ) @ ( real_V1485227260804924795R_real @ B3 @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_492_scaleR__right__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ B3 @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_493_scaleR__left__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( real_V1485227260804924795R_real @ C @ B3 ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_494_scaleR__left__mono,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ A3 @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_495_scaleR__nonpos__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A3 @ B3 ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_496_scaleR__nonpos__nonneg,axiom,
! [A3: real,X: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_497_complete__interval,axiom,
! [A3: real,B3: real,P: real > $o] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A3 @ C3 )
& ( ord_less_eq_real @ C3 @ B3 )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_real @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A3 @ X2 )
& ( ord_less_real @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_498_complete__interval,axiom,
! [A3: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A3 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A3 @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A3 @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_499_pinf_I6_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_500_pinf_I6_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_501_pinf_I8_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_eq_real @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_502_pinf_I8_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_503_minf_I6_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ord_less_eq_real @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_504_minf_I6_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_505_minf_I8_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ~ ( ord_less_eq_real @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_506_minf_I8_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ~ ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_507_scaleR__left__le__one__le,axiom,
! [X: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_508_sum__pos2,axiom,
! [I4: set_set_nat,I2: set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5107569545109728110t_real @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_509_sum__pos2,axiom,
! [I4: set_real,I2: real,F: real > nat] :
( ( finite_finite_real @ I4 )
=> ( ( member_real @ I2 @ I4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_510_sum__pos2,axiom,
! [I4: set_set_nat,I2: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups8294997508430121362at_nat @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_511_sum__pos2,axiom,
! [I4: set_nat,I2: nat,F: nat > nat] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_512_sum__pos2,axiom,
! [I4: set_nat,I2: nat,F: nat > real] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_513_sum__pos2,axiom,
! [I4: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I4 )
=> ( ( member_real @ I2 @ I4 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_514_pos__divideR__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) @ A3 )
= ( ord_less_real @ B3 @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_515_pos__less__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A3 @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ B3 ) ) ) ).
% pos_less_divideR_eq
thf(fact_516_neg__divideR__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) @ A3 )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ B3 ) ) ) ).
% neg_divideR_less_eq
thf(fact_517_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_518_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_519_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > risk_Free_account] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_520_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_521_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_522_sum_Oinfinite,axiom,
! [A: set_real,G: real > real] :
( ~ ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_523_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_524_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_525_sum_Odelta_H,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > real] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups5107569545109728110t_real
@ ^ [K3: set_nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups5107569545109728110t_real
@ ^ [K3: set_nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_526_sum_Odelta_H,axiom,
! [S: set_real,A3: real,B3: real > risk_Free_account] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_527_sum_Odelta_H,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups5807469391267537845ccount
@ ^ [K3: set_nat] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups5807469391267537845ccount
@ ^ [K3: set_nat] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_528_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > risk_Free_account] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta'
thf(fact_529_sum_Odelta_H,axiom,
! [S: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_530_sum_Odelta_H,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups8294997508430121362at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups8294997508430121362at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_531_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_532_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_533_sum_Odelta_H,axiom,
! [S: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_534_sum_Odelta,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > real] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups5107569545109728110t_real
@ ^ [K3: set_nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups5107569545109728110t_real
@ ^ [K3: set_nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_535_sum_Odelta,axiom,
! [S: set_real,A3: real,B3: real > risk_Free_account] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups8516999891779824987ccount
@ ^ [K3: real] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_536_sum_Odelta,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups5807469391267537845ccount
@ ^ [K3: set_nat] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups5807469391267537845ccount
@ ^ [K3: set_nat] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_537_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > risk_Free_account] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( if_Risk_Free_account @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_z1425366712893667068ccount )
@ S )
= zero_z1425366712893667068ccount ) ) ) ) ).
% sum.delta
thf(fact_538_sum_Odelta,axiom,
! [S: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_539_sum_Odelta,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups8294997508430121362at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups8294997508430121362at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_540_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_541_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_542_sum_Odelta,axiom,
! [S: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_543_norm__inverse,axiom,
! [A3: complex] :
( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A3 ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A3 ) ) ) ).
% norm_inverse
thf(fact_544_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_545_nonzero__norm__inverse,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A3 ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A3 ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_546_nonzero__norm__inverse,axiom,
! [A3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A3 ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A3 ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_547_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_548_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_549_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_550_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_551_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_552_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_553_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_554_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_555_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_556_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_557_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_558_infinite__Iic,axiom,
! [A3: real] :
~ ( finite_finite_real @ ( set_ord_atMost_real @ A3 ) ) ).
% infinite_Iic
thf(fact_559_nonzero__inverse__scaleR__distrib,axiom,
! [A3: real,X: real] :
( ( A3 != zero_zero_real )
=> ( ( X != zero_zero_real )
=> ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A3 @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_560_sum_Oswap__restrict,axiom,
! [A: set_set_nat,B: set_nat,G: set_nat > nat > real,R: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5107569545109728110t_real
@ ^ [X4: set_nat] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups5107569545109728110t_real
@ ^ [X4: set_nat] : ( G @ X4 @ Y4 )
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_561_sum_Oswap__restrict,axiom,
! [A: set_set_nat,B: set_real,G: set_nat > real > real,R: set_nat > real > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups5107569545109728110t_real
@ ^ [X4: set_nat] :
( groups8097168146408367636l_real @ ( G @ X4 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8097168146408367636l_real
@ ^ [Y4: real] :
( groups5107569545109728110t_real
@ ^ [X4: set_nat] : ( G @ X4 @ Y4 )
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_562_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_set_nat,G: nat > set_nat > real,R: nat > set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( groups6591440286371151544t_real
@ ^ [X4: nat] :
( groups5107569545109728110t_real @ ( G @ X4 )
@ ( collect_set_nat
@ ^ [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups5107569545109728110t_real
@ ^ [Y4: set_nat] :
( groups6591440286371151544t_real
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_563_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_nat,G: nat > nat > real,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups6591440286371151544t_real
@ ^ [X4: nat] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups6591440286371151544t_real
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_564_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_real,G: nat > real > real,R: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups6591440286371151544t_real
@ ^ [X4: nat] :
( groups8097168146408367636l_real @ ( G @ X4 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8097168146408367636l_real
@ ^ [Y4: real] :
( groups6591440286371151544t_real
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_565_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_set_nat,G: real > set_nat > real,R: real > set_nat > $o] :
( ( finite_finite_real @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( groups8097168146408367636l_real
@ ^ [X4: real] :
( groups5107569545109728110t_real @ ( G @ X4 )
@ ( collect_set_nat
@ ^ [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups5107569545109728110t_real
@ ^ [Y4: set_nat] :
( groups8097168146408367636l_real
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_566_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
@ ^ [X4: real] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8097168146408367636l_real
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_567_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_real,G: real > real > real,R: real > real > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups8097168146408367636l_real
@ ^ [X4: real] :
( groups8097168146408367636l_real @ ( G @ X4 )
@ ( collect_real
@ ^ [Y4: real] :
( ( member_real @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8097168146408367636l_real
@ ^ [Y4: real] :
( groups8097168146408367636l_real
@ ^ [X4: real] : ( G @ X4 @ Y4 )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_568_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_569_norm__inverse__le__norm,axiom,
! [R2: real,X: complex] :
( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_570_sum__mono__inv,axiom,
! [F: set_nat > real,I4: set_set_nat,G: set_nat > real,I2: set_nat] :
( ( ( groups5107569545109728110t_real @ F @ I4 )
= ( groups5107569545109728110t_real @ G @ I4 ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( finite1152437895449049373et_nat @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_571_sum__mono__inv,axiom,
! [F: real > nat,I4: set_real,G: real > nat,I2: 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 @ I2 @ I4 )
=> ( ( finite_finite_real @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_572_sum__mono__inv,axiom,
! [F: set_nat > nat,I4: set_set_nat,G: set_nat > nat,I2: set_nat] :
( ( ( groups8294997508430121362at_nat @ F @ I4 )
= ( groups8294997508430121362at_nat @ G @ I4 ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( finite1152437895449049373et_nat @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_573_sum__mono__inv,axiom,
! [F: nat > nat,I4: set_nat,G: nat > nat,I2: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I4 )
= ( groups3542108847815614940at_nat @ G @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_574_sum__mono__inv,axiom,
! [F: nat > real,I4: set_nat,G: nat > real,I2: nat] :
( ( ( groups6591440286371151544t_real @ F @ I4 )
= ( groups6591440286371151544t_real @ G @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_575_sum__mono__inv,axiom,
! [F: real > real,I4: set_real,G: real > real,I2: 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 @ I2 @ I4 )
=> ( ( finite_finite_real @ I4 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_576_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_577_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_578_sum_Ointer__filter,axiom,
! [A: set_set_nat,G: set_nat > real,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( groups5107569545109728110t_real @ G
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups5107569545109728110t_real
@ ^ [X4: set_nat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_579_sum_Ointer__filter,axiom,
! [A: set_real,G: real > risk_Free_account,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8516999891779824987ccount @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8516999891779824987ccount
@ ^ [X4: real] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_580_sum_Ointer__filter,axiom,
! [A: set_set_nat,G: set_nat > risk_Free_account,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( groups5807469391267537845ccount @ G
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups5807469391267537845ccount
@ ^ [X4: set_nat] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_581_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > risk_Free_account,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups6033208628184776703ccount
@ ^ [X4: nat] : ( if_Risk_Free_account @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z1425366712893667068ccount )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_582_sum_Ointer__filter,axiom,
! [A: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X4: real] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_583_sum_Ointer__filter,axiom,
! [A: set_set_nat,G: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( groups8294997508430121362at_nat @ G
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8294997508430121362at_nat
@ ^ [X4: set_nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_584_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_585_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_586_sum_Ointer__filter,axiom,
! [A: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_587_sum__le__included,axiom,
! [S2: set_nat,T3: set_nat,G: nat > nat,I2: 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 )
& ( ( I2 @ 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_588_sum__le__included,axiom,
! [S2: set_nat,T3: set_nat,G: nat > real,I2: nat > nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I2 @ Xa )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ S2 ) @ ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_589_sum__le__included,axiom,
! [S2: set_nat,T3: set_real,G: real > real,I2: real > nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ T3 )
& ( ( I2 @ Xa )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ S2 ) @ ( groups8097168146408367636l_real @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_590_sum__le__included,axiom,
! [S2: set_real,T3: set_nat,G: nat > real,I2: nat > real,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T3 )
& ( ( I2 @ Xa )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ S2 ) @ ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_591_sum__le__included,axiom,
! [S2: set_real,T3: set_real,G: real > real,I2: real > real,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ T3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ T3 )
& ( ( I2 @ Xa )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ S2 ) @ ( groups8097168146408367636l_real @ G @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_592_sum__nonneg__eq__0__iff,axiom,
! [A: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups5107569545109728110t_real @ F @ A )
= zero_zero_real )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_593_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 )
= ( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_594_sum__nonneg__eq__0__iff,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_595_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_596_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > real] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ A )
= zero_zero_real )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_597_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 )
= ( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_598_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_599_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_600_sum__strict__mono__ex1,axiom,
! [A: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_601_minf_I7_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ~ ( ord_less_real @ T3 @ X3 ) ) ).
% minf(7)
thf(fact_602_minf_I7_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ~ ( ord_less_nat @ T3 @ X3 ) ) ).
% minf(7)
thf(fact_603_minf_I5_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ord_less_real @ X3 @ T3 ) ) ).
% minf(5)
thf(fact_604_minf_I5_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ord_less_nat @ X3 @ T3 ) ) ).
% minf(5)
thf(fact_605_minf_I4_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(4)
thf(fact_606_minf_I4_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(4)
thf(fact_607_minf_I3_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(3)
thf(fact_608_minf_I3_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(3)
thf(fact_609_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_610_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_611_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_612_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_613_pinf_I7_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_real @ T3 @ X3 ) ) ).
% pinf(7)
thf(fact_614_pinf_I7_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ord_less_nat @ T3 @ X3 ) ) ).
% pinf(7)
thf(fact_615_pinf_I5_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_real @ X3 @ T3 ) ) ).
% pinf(5)
thf(fact_616_pinf_I5_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T3 ) ) ).
% pinf(5)
thf(fact_617_pinf_I4_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(4)
thf(fact_618_pinf_I4_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(4)
thf(fact_619_pinf_I3_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(3)
thf(fact_620_pinf_I3_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(3)
thf(fact_621_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_622_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_623_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_624_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_625_ex__gt__or__lt,axiom,
! [A3: real] :
? [B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( ord_less_real @ B2 @ A3 ) ) ).
% ex_gt_or_lt
thf(fact_626_sum__nonneg__0,axiom,
! [S2: set_set_nat,F: set_nat > real,I2: set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5107569545109728110t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_set_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_627_sum__nonneg__0,axiom,
! [S2: set_real,F: real > nat,I2: 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 @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_628_sum__nonneg__0,axiom,
! [S2: set_set_nat,F: set_nat > nat,I2: set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_set_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_629_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > nat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_630_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > real,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_631_sum__nonneg__0,axiom,
! [S2: set_real,F: real > real,I2: 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 @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_632_sum__nonneg__leq__bound,axiom,
! [S2: set_set_nat,F: set_nat > real,B: real,I2: set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5107569545109728110t_real @ F @ S2 )
= B )
=> ( ( member_set_nat @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_633_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > nat,B: nat,I2: 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 @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_634_sum__nonneg__leq__bound,axiom,
! [S2: set_set_nat,F: set_nat > nat,B: nat,I2: set_nat] :
( ( finite1152437895449049373et_nat @ S2 )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups8294997508430121362at_nat @ F @ S2 )
= B )
=> ( ( member_set_nat @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_635_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > nat,B: nat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= B )
=> ( ( member_nat @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_636_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > real,B: real,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S2 )
= B )
=> ( ( member_nat @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_637_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > real,B: real,I2: 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 @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_638_neg__le__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) )
= ( ord_less_eq_real @ B3 @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_639_neg__divideR__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) @ A3 )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ B3 ) ) ) ).
% neg_divideR_le_eq
thf(fact_640_pos__le__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A3 @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ B3 ) ) ) ).
% pos_le_divideR_eq
thf(fact_641_pos__divideR__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) @ A3 )
= ( ord_less_eq_real @ B3 @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_642_neg__less__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) )
= ( ord_less_real @ B3 @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_643_inverse__le__iff__le__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_644_inverse__le__iff__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_645_inverse__less__iff__less,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_646_inverse__less__iff__less__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_647_inverse__negative__iff__negative,axiom,
! [A3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_648_inverse__positive__iff__positive,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% inverse_positive_iff_positive
thf(fact_649_inverse__nonpositive__iff__nonpositive,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_650_inverse__nonzero__iff__nonzero,axiom,
! [A3: real] :
( ( ( inverse_inverse_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_651_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_652_inverse__nonnegative__iff__nonnegative,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_653_psubsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% psubsetD
thf(fact_654_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_655_psubsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_656_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_657_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_658_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_659_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_660_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_661_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_662_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_663_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_664_sum__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= one_one_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_665_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_666_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X3 ) ).
% linordered_field_no_lb
thf(fact_667_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_668_inverse__zero__imp__zero,axiom,
! [A3: real] :
( ( ( inverse_inverse_real @ A3 )
= zero_zero_real )
=> ( A3 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_669_nonzero__inverse__eq__imp__eq,axiom,
! [A3: real,B3: real] :
( ( ( inverse_inverse_real @ A3 )
= ( inverse_inverse_real @ B3 ) )
=> ( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( A3 = B3 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_670_nonzero__inverse__inverse__eq,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A3 ) )
= A3 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_671_nonzero__imp__inverse__nonzero,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( inverse_inverse_real @ A3 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_672_positive__imp__inverse__positive,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A3 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_673_negative__imp__inverse__negative,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_674_inverse__positive__imp__positive,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A3 ) )
=> ( ( A3 != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A3 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_675_inverse__negative__imp__negative,axiom,
! [A3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_676_less__imp__inverse__less__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_677_inverse__less__imp__less__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_678_less__imp__inverse__less,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_679_inverse__less__imp__less,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ B3 @ A3 ) ) ) ).
% inverse_less_imp_less
thf(fact_680_inverse__le__imp__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% inverse_le_imp_le
thf(fact_681_le__imp__inverse__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_682_inverse__le__imp__le__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A3 ) @ ( inverse_inverse_real @ B3 ) )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_683_le__imp__inverse__le__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_684_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_685_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_686_one__less__inverse,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ A3 @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% one_less_inverse
thf(fact_687_one__le__inverse,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% one_le_inverse
thf(fact_688_inverse__less__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_689_one__le__inverse__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_690_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_691_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_692_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_693_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_694_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_695_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_696_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_697_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_698_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_699_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
@ ^ [A4: real] :
( ( member_real @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_700_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_701_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
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_702_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_703_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_704_finite__has__maximal2,axiom,
! [A: set_real,A3: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A3 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ A3 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_705_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_706_finite__has__maximal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A3 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_707_finite__has__minimal2,axiom,
! [A: set_real,A3: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A3 @ A )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( ord_less_eq_real @ X2 @ A3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_708_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_709_finite__has__minimal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_710_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_711_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_712_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_713_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P @ B7 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_714_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_715_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_716_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_717_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_718_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_719_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_720_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_721_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_722_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_723_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_724_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_725_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_726_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_727_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_728_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_729_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_730_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_731_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_732_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_733_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( P @ M3 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_734_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_735_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_736_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_737_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_738_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_739_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_740_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_741_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_742_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_743_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_744_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_745_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_746_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_747_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_748_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N2 @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K4 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K4 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_749_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_750_independentD,axiom,
! [S2: set_real,T3: set_real,U2: real > real,V: real] :
( ~ ( real_V7051607973971999986t_real @ S2 )
=> ( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ T3 @ S2 )
=> ( ( ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( U2 @ V2 ) @ V2 )
@ T3 )
= zero_zero_real )
=> ( ( member_real @ V @ T3 )
=> ( ( U2 @ V )
= zero_zero_real ) ) ) ) ) ) ).
% independentD
thf(fact_751_dependent__alt,axiom,
( real_V7051607973971999986t_real
= ( ^ [B5: set_real] :
? [X7: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) ) )
& ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) )
@ B5 )
& ( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( X7 @ X4 ) @ X4 )
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) ) )
= zero_zero_real )
& ? [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) ) ) ) ).
% dependent_alt
thf(fact_752_dependent__zero,axiom,
! [A: set_real] :
( ( member_real @ zero_zero_real @ A )
=> ( real_V7051607973971999986t_real @ A ) ) ).
% dependent_zero
thf(fact_753_unique__representation,axiom,
! [Basis: set_real,F: real > real,G: real > real] :
( ~ ( real_V7051607973971999986t_real @ Basis )
=> ( ! [V3: real] :
( ( ( F @ V3 )
!= zero_zero_real )
=> ( member_real @ V3 @ Basis ) )
=> ( ! [V3: real] :
( ( ( G @ V3 )
!= zero_zero_real )
=> ( member_real @ V3 @ Basis ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [V2: real] :
( ( F @ V2 )
!= zero_zero_real ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [V2: real] :
( ( G @ V2 )
!= zero_zero_real ) ) )
=> ( ( ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( F @ V2 ) @ V2 )
@ ( collect_real
@ ^ [V2: real] :
( ( F @ V2 )
!= zero_zero_real ) ) )
= ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( G @ V2 ) @ V2 )
@ ( collect_real
@ ^ [V2: real] :
( ( G @ V2 )
!= zero_zero_real ) ) ) )
=> ( F = G ) ) ) ) ) ) ) ).
% unique_representation
thf(fact_754_independent__if__scalars__zero,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ! [F3: real > real,X2: real] :
( ( ( groups8097168146408367636l_real
@ ^ [Y4: real] : ( real_V1485227260804924795R_real @ ( F3 @ Y4 ) @ Y4 )
@ A )
= zero_zero_real )
=> ( ( member_real @ X2 @ A )
=> ( ( F3 @ X2 )
= zero_zero_real ) ) )
=> ~ ( real_V7051607973971999986t_real @ A ) ) ) ).
% independent_if_scalars_zero
thf(fact_755_dependent__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( real_V7051607973971999986t_real @ S )
= ( ? [U: real > real] :
( ? [X4: real] :
( ( member_real @ X4 @ S )
& ( ( U @ X4 )
!= zero_zero_real ) )
& ( ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( U @ V2 ) @ V2 )
@ S )
= zero_zero_real ) ) ) ) ) ).
% dependent_finite
thf(fact_756_independentD__unique,axiom,
! [B: set_real,X5: real > real,Y6: real > real] :
( ~ ( real_V7051607973971999986t_real @ B )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) ) )
=> ( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) )
@ B )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( Y6 @ X4 )
!= zero_zero_real ) ) )
=> ( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( Y6 @ X4 )
!= zero_zero_real ) )
@ B )
=> ( ( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( X5 @ X4 ) @ X4 )
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( Y6 @ X4 ) @ X4 )
@ ( collect_real
@ ^ [X4: real] :
( ( Y6 @ X4 )
!= zero_zero_real ) ) ) )
=> ( X5 = Y6 ) ) ) ) ) ) ) ).
% independentD_unique
thf(fact_757_independent__explicit__finite__subsets,axiom,
! [A: set_real] :
( ( ~ ( real_V7051607973971999986t_real @ A ) )
= ( ! [S3: set_real] :
( ( ord_less_eq_set_real @ S3 @ A )
=> ( ( finite_finite_real @ S3 )
=> ! [U: real > real] :
( ( ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( U @ V2 ) @ V2 )
@ S3 )
= zero_zero_real )
=> ! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ( ( U @ X4 )
= zero_zero_real ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_758_independent__explicit__module,axiom,
! [S2: set_real] :
( ( ~ ( real_V7051607973971999986t_real @ S2 ) )
= ( ! [T2: set_real,U: real > real,V2: real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ T2 @ S2 )
=> ( ( ( groups8097168146408367636l_real
@ ^ [W2: real] : ( real_V1485227260804924795R_real @ ( U @ W2 ) @ W2 )
@ T2 )
= zero_zero_real )
=> ( ( member_real @ V2 @ T2 )
=> ( ( U @ V2 )
= zero_zero_real ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_759_dependent__explicit,axiom,
( real_V7051607973971999986t_real
= ( ^ [S4: set_real] :
? [T2: set_real] :
( ( finite_finite_real @ T2 )
& ( ord_less_eq_set_real @ T2 @ S4 )
& ? [U: real > real] :
( ( ( groups8097168146408367636l_real
@ ^ [V2: real] : ( real_V1485227260804924795R_real @ ( U @ V2 ) @ V2 )
@ T2 )
= zero_zero_real )
& ? [X4: real] :
( ( member_real @ X4 @ T2 )
& ( ( U @ X4 )
!= zero_zero_real ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_760_independentD__alt,axiom,
! [B: set_real,X5: real > real,X: real] :
( ~ ( real_V7051607973971999986t_real @ B )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) ) )
=> ( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) )
@ B )
=> ( ( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( X5 @ X4 ) @ X4 )
@ ( collect_real
@ ^ [X4: real] :
( ( X5 @ X4 )
!= zero_zero_real ) ) )
= zero_zero_real )
=> ( ( X5 @ X )
= zero_zero_real ) ) ) ) ) ).
% independentD_alt
thf(fact_761_independent__alt,axiom,
! [B: set_real] :
( ( ~ ( real_V7051607973971999986t_real @ B ) )
= ( ! [X7: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) ) )
=> ( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) )
@ B )
=> ( ( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( X7 @ X4 ) @ X4 )
@ ( collect_real
@ ^ [X4: real] :
( ( X7 @ X4 )
!= zero_zero_real ) ) )
= zero_zero_real )
=> ! [X4: real] :
( ( X7 @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% independent_alt
thf(fact_762_sum__strict__mono2,axiom,
! [B: set_set_nat,A: set_set_nat,B3: set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_real @ ( groups5107569545109728110t_real @ F @ A ) @ ( groups5107569545109728110t_real @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_763_sum__strict__mono2,axiom,
! [B: set_real,A: set_real,B3: real,F: real > nat] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_764_sum__strict__mono2,axiom,
! [B: set_set_nat,A: set_set_nat,B3: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups8294997508430121362at_nat @ F @ A ) @ ( groups8294997508430121362at_nat @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_765_sum__strict__mono2,axiom,
! [B: set_nat,A: set_nat,B3: nat,F: nat > nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_766_sum__strict__mono2,axiom,
! [B: set_nat,A: set_nat,B3: nat,F: nat > real] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_767_sum__strict__mono2,axiom,
! [B: set_real,A: set_real,B3: real,F: real > real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ B3 @ ( minus_minus_set_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ F @ B ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_768_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_769_neg__minus__divideR__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) @ A3 )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_770_neg__less__minus__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_771_pos__minus__divideR__less__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) @ A3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_772_add_Oinverse__inverse,axiom,
! [A3: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_773_add_Oinverse__inverse,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_774_neg__equal__iff__equal,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A3 )
= ( uminus3377898441596595772ccount @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_775_neg__equal__iff__equal,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B3 ) )
= ( A3 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_776_verit__minus__simplify_I4_J,axiom,
! [B3: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_777_verit__minus__simplify_I4_J,axiom,
! [B3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_778_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_779_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_780_DiffI,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ A )
=> ( ~ ( member_real @ C @ B )
=> ( member_real @ C @ ( minus_minus_set_real @ A @ B ) ) ) ) ).
% DiffI
thf(fact_781_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_782_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_783_Diff__iff,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
= ( ( member_real @ C @ A )
& ~ ( member_real @ C @ B ) ) ) ).
% Diff_iff
thf(fact_784_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_785_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_786_Rep__account__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N ) ) ) ) ).
% Rep_account_uminus
thf(fact_787_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_788_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_789_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ A3 )
= zero_z1425366712893667068ccount ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_790_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_791_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_792_diff__zero,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% diff_zero
thf(fact_793_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_794_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_795_diff__0__right,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% diff_0_right
thf(fact_796_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_797_diff__self,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ A3 )
= zero_z1425366712893667068ccount ) ).
% diff_self
thf(fact_798_neg__le__iff__le,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_799_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_800_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_801_neg__equal__0__iff__equal,axiom,
! [A3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A3 )
= zero_z1425366712893667068ccount )
= ( A3 = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_802_neg__equal__0__iff__equal,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_803_neg__0__equal__iff__equal,axiom,
! [A3: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A3 ) )
= ( zero_z1425366712893667068ccount = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_804_neg__0__equal__iff__equal,axiom,
! [A3: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A3 ) )
= ( zero_zero_real = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_805_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_806_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_807_neg__less__iff__less,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_808_minus__diff__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) )
= ( minus_4846202936726426316ccount @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_809_minus__diff__eq,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% minus_diff_eq
thf(fact_810_norm__minus__cancel,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ X ) )
= ( real_V7735802525324610683m_real @ X ) ) ).
% norm_minus_cancel
thf(fact_811_norm__minus__cancel,axiom,
! [X: complex] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ X ) )
= ( real_V1022390504157884413omplex @ X ) ) ).
% norm_minus_cancel
thf(fact_812_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_813_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_814_scaleR__minus__left,axiom,
! [A3: real,X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ A3 ) @ X )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) ) ) ).
% scaleR_minus_left
thf(fact_815_scaleR__left_Ominus,axiom,
! [X: real,Xa2: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ X ) @ Xa2 )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) ) ) ).
% scaleR_left.minus
thf(fact_816_inverse__minus__eq,axiom,
! [A3: real] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ A3 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A3 ) ) ) ).
% inverse_minus_eq
thf(fact_817_scaleR__minus__right,axiom,
! [A3: real,X: real] :
( ( real_V1485227260804924795R_real @ A3 @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) ) ) ).
% scaleR_minus_right
thf(fact_818_of__real__diff,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_diff
thf(fact_819_of__real__minus,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_minus
thf(fact_820_minus__of__real__eq__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) )
= ( real_V1803761363581548252l_real @ Y ) )
= ( ( uminus_uminus_real @ X )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_821_of__real__eq__minus__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V1803761363581548252l_real @ X )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ Y ) ) )
= ( X
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_822_diff__ge__0__iff__ge,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_823_neg__0__le__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_824_neg__le__0__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_825_less__eq__neg__nonpos,axiom,
! [A3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_826_neg__less__eq__nonneg,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_827_diff__gt__0__iff__gt,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_828_less__neg__neg,axiom,
! [A3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_829_neg__less__pos,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_pos
thf(fact_830_neg__0__less__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_831_neg__less__0__iff__less,axiom,
! [A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% neg_less_0_iff_less
thf(fact_832_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_833_verit__minus__simplify_I3_J,axiom,
! [B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ B3 )
= ( uminus3377898441596595772ccount @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_834_verit__minus__simplify_I3_J,axiom,
! [B3: real] :
( ( minus_minus_real @ zero_zero_real @ B3 )
= ( uminus_uminus_real @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_835_diff__0,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ A3 )
= ( uminus3377898441596595772ccount @ A3 ) ) ).
% diff_0
thf(fact_836_diff__0,axiom,
! [A3: real] :
( ( minus_minus_real @ zero_zero_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ).
% diff_0
thf(fact_837_scaleR__minus1__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus_uminus_real @ X ) ) ).
% scaleR_minus1_left
thf(fact_838_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_839_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_840_diff__mono,axiom,
! [A3: real,B3: real,D: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_841_diff__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% diff_left_mono
thf(fact_842_diff__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% diff_right_mono
thf(fact_843_diff__eq__diff__less__eq,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A3 @ B3 )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_844_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_845_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [A4: risk_Free_account,B4: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A4 @ B4 )
= zero_z1425366712893667068ccount ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_846_diff__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_847_diff__strict__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_848_diff__eq__diff__less,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A3 @ B3 )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_849_diff__strict__mono,axiom,
! [A3: real,B3: real,D: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_850_le__minus__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_minus_iff
thf(fact_851_minus__le__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% minus_le_iff
thf(fact_852_le__imp__neg__le,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% le_imp_neg_le
thf(fact_853_verit__negate__coefficient_I2_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_854_minus__less__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_855_less__minus__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% less_minus_iff
thf(fact_856_norm__triangle__ineq2,axiom,
! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq2
thf(fact_857_norm__triangle__ineq2,axiom,
! [A3: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ B3 ) ) ) ).
% norm_triangle_ineq2
thf(fact_858_set__diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B5: set_real] :
( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A5 )
& ~ ( member_real @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_859_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A5 )
& ~ ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_860_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A5 )
& ~ ( member_nat @ X4 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_861_diff__eq__diff__eq,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_862_diff__eq__diff__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A3 @ B3 )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_863_equation__minus__iff,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( A3
= ( uminus3377898441596595772ccount @ B3 ) )
= ( B3
= ( uminus3377898441596595772ccount @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_864_equation__minus__iff,axiom,
! [A3: real,B3: real] :
( ( A3
= ( uminus_uminus_real @ B3 ) )
= ( B3
= ( uminus_uminus_real @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_865_minus__equation__iff,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A3 )
= B3 )
= ( ( uminus3377898441596595772ccount @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_866_minus__equation__iff,axiom,
! [A3: real,B3: real] :
( ( ( uminus_uminus_real @ A3 )
= B3 )
= ( ( uminus_uminus_real @ B3 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_867_minus__diff__commute,axiom,
! [B3: risk_Free_account,A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ B3 ) @ A3 )
= ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_868_minus__diff__commute,axiom,
! [B3: real,A3: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A3 )
= ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% minus_diff_commute
thf(fact_869_diff__right__commute,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_870_diff__right__commute,axiom,
! [A3: real,C: real,B3: real] :
( ( minus_minus_real @ ( minus_minus_real @ A3 @ C ) @ B3 )
= ( minus_minus_real @ ( minus_minus_real @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_871_diff__right__commute,axiom,
! [A3: risk_Free_account,C: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A3 @ C ) @ B3 )
= ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_872_DiffE,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% DiffE
thf(fact_873_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_874_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_875_DiffD1,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ( member_real @ C @ A ) ) ).
% DiffD1
thf(fact_876_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_877_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_878_DiffD2,axiom,
! [C: real,A: set_real,B: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( member_real @ C @ B ) ) ).
% DiffD2
thf(fact_879_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_880_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_881_verit__negate__coefficient_I3_J,axiom,
! [A3: real,B3: real] :
( ( A3 = B3 )
=> ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_882_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A3: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_883_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A3: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_884_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A3: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_885_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A3: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_886_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A3: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_887_minus__diff__minus,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ A3 ) @ ( uminus3377898441596595772ccount @ B3 ) )
= ( uminus3377898441596595772ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) ) ) ).
% minus_diff_minus
thf(fact_888_minus__diff__minus,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).
% minus_diff_minus
thf(fact_889_scaleR__left__diff__distrib,axiom,
! [A3: real,B3: real,X: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A3 @ B3 ) @ X )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ B3 @ X ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_890_scaleR__left_Odiff,axiom,
! [X: real,Y: real,Xa2: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa2 )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y @ Xa2 ) ) ) ).
% scaleR_left.diff
thf(fact_891_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_892_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_893_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D4: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D4 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_894_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_895_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_896_norm__minus__commute,axiom,
! [A3: real,B3: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( real_V7735802525324610683m_real @ ( minus_minus_real @ B3 @ A3 ) ) ) ).
% norm_minus_commute
thf(fact_897_norm__minus__commute,axiom,
! [A3: complex,B3: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A3 @ B3 ) )
= ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B3 @ A3 ) ) ) ).
% norm_minus_commute
thf(fact_898_scaleR__right__diff__distrib,axiom,
! [A3: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A3 @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ A3 @ Y ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_899_psubset__imp__ex__mem,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_set_real @ A @ B )
=> ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_900_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_901_psubset__imp__ex__mem,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ? [B2: set_nat] : ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_902_sum__diff,axiom,
! [A: set_nat,B: set_nat,F: nat > risk_Free_account] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( groups6033208628184776703ccount @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_4846202936726426316ccount @ ( groups6033208628184776703ccount @ F @ A ) @ ( groups6033208628184776703ccount @ F @ B ) ) ) ) ) ).
% sum_diff
thf(fact_903_sum__diff,axiom,
! [A: set_nat,B: set_nat,F: nat > real] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ F @ B ) ) ) ) ) ).
% sum_diff
thf(fact_904_sum__diff,axiom,
! [A: set_real,B: set_real,F: real > real] :
( ( finite_finite_real @ A )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A @ B ) )
= ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ F @ B ) ) ) ) ) ).
% sum_diff
thf(fact_905_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N ) ) ) ) ) ).
% uminus_account_def
thf(fact_906_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( minus_minus_real @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_907_sum__subtractf,axiom,
! [F: real > real,G: real > real,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( minus_minus_real @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_908_sum__negf,axiom,
! [F: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( uminus_uminus_real @ ( F @ X4 ) )
@ A )
= ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_909_sum__negf,axiom,
! [F: real > real,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( uminus_uminus_real @ ( F @ X4 ) )
@ A )
= ( uminus_uminus_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_910_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_911_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_912_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_913_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_914_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_915_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_916_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_917_nonzero__inverse__minus__eq,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A3 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A3 ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_918_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_919_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_920_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_921_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_922_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S: set_real,I2: real > real,J2: real > real,T: set_real,G: real > risk_Free_account,H: real > risk_Free_account] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups8516999891779824987ccount @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_923_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_nat,S: set_real,I2: nat > real,J2: real > nat,T: set_nat,G: real > risk_Free_account,H: nat > risk_Free_account] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups6033208628184776703ccount @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_924_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_real,S: set_nat,I2: real > nat,J2: nat > real,T: set_real,G: nat > risk_Free_account,H: real > risk_Free_account] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups8516999891779824987ccount @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_925_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_nat,S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups6033208628184776703ccount @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_926_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S: set_real,I2: real > real,J2: real > real,T: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_927_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_nat,S: set_real,I2: nat > real,J2: real > nat,T: set_nat,G: real > nat,H: nat > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_928_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_real,S: set_nat,I2: real > nat,J2: nat > real,T: set_real,G: nat > nat,H: real > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_929_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_nat,S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_930_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_nat,S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_931_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_real,S: set_nat,I2: real > nat,J2: nat > real,T: set_real,G: nat > real,H: real > real] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_932_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > risk_Free_account] :
( ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= zero_z1425366712893667068ccount ) ) ) )
= ( groups6033208628184776703ccount @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_933_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_934_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > real] :
( ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= zero_zero_real ) ) ) )
= ( groups6591440286371151544t_real @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_935_sum_Osetdiff__irrelevant,axiom,
! [A: set_real,G: real > real] :
( ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G
@ ( minus_minus_set_real @ A
@ ( collect_real
@ ^ [X4: real] :
( ( G @ X4 )
= zero_zero_real ) ) ) )
= ( groups8097168146408367636l_real @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_936_sum_Osame__carrier,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > real,H: set_nat > real] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5107569545109728110t_real @ G @ A )
= ( groups5107569545109728110t_real @ H @ B ) )
= ( ( groups5107569545109728110t_real @ G @ C2 )
= ( groups5107569545109728110t_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_937_sum_Osame__carrier,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > risk_Free_account,H: real > risk_Free_account] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups8516999891779824987ccount @ G @ A )
= ( groups8516999891779824987ccount @ H @ B ) )
= ( ( groups8516999891779824987ccount @ G @ C2 )
= ( groups8516999891779824987ccount @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_938_sum_Osame__carrier,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > risk_Free_account,H: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups5807469391267537845ccount @ G @ A )
= ( groups5807469391267537845ccount @ H @ B ) )
= ( ( groups5807469391267537845ccount @ G @ C2 )
= ( groups5807469391267537845ccount @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_939_sum_Osame__carrier,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A )
= ( groups1935376822645274424al_nat @ H @ B ) )
= ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_940_sum_Osame__carrier,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups8294997508430121362at_nat @ G @ A )
= ( groups8294997508430121362at_nat @ H @ B ) )
= ( ( groups8294997508430121362at_nat @ G @ C2 )
= ( groups8294997508430121362at_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_941_sum_Osame__carrier,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups6033208628184776703ccount @ G @ A )
= ( groups6033208628184776703ccount @ H @ B ) )
= ( ( groups6033208628184776703ccount @ G @ C2 )
= ( groups6033208628184776703ccount @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_942_sum_Osame__carrier,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B ) )
= ( ( groups3542108847815614940at_nat @ G @ C2 )
= ( groups3542108847815614940at_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_943_sum_Osame__carrier,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) )
= ( ( groups6591440286371151544t_real @ G @ C2 )
= ( groups6591440286371151544t_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_944_sum_Osame__carrier,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A )
= ( groups8097168146408367636l_real @ H @ B ) )
= ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_945_sum_Osame__carrierI,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > real,H: set_nat > real] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups5107569545109728110t_real @ G @ C2 )
= ( groups5107569545109728110t_real @ H @ C2 ) )
=> ( ( groups5107569545109728110t_real @ G @ A )
= ( groups5107569545109728110t_real @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_946_sum_Osame__carrierI,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > risk_Free_account,H: real > risk_Free_account] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups8516999891779824987ccount @ G @ C2 )
= ( groups8516999891779824987ccount @ H @ C2 ) )
=> ( ( groups8516999891779824987ccount @ G @ A )
= ( groups8516999891779824987ccount @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_947_sum_Osame__carrierI,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > risk_Free_account,H: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups5807469391267537845ccount @ G @ C2 )
= ( groups5807469391267537845ccount @ H @ C2 ) )
=> ( ( groups5807469391267537845ccount @ G @ A )
= ( groups5807469391267537845ccount @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_948_sum_Osame__carrierI,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A )
= ( groups1935376822645274424al_nat @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_949_sum_Osame__carrierI,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups8294997508430121362at_nat @ G @ C2 )
= ( groups8294997508430121362at_nat @ H @ C2 ) )
=> ( ( groups8294997508430121362at_nat @ G @ A )
= ( groups8294997508430121362at_nat @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_950_sum_Osame__carrierI,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_z1425366712893667068ccount ) )
=> ( ( ( groups6033208628184776703ccount @ G @ C2 )
= ( groups6033208628184776703ccount @ H @ C2 ) )
=> ( ( groups6033208628184776703ccount @ G @ A )
= ( groups6033208628184776703ccount @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_951_sum_Osame__carrierI,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ C2 )
= ( groups3542108847815614940at_nat @ H @ C2 ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_952_sum_Osame__carrierI,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups6591440286371151544t_real @ G @ C2 )
= ( groups6591440286371151544t_real @ H @ C2 ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_953_sum_Osame__carrierI,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= zero_zero_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) )
=> ( ( groups8097168146408367636l_real @ G @ A )
= ( groups8097168146408367636l_real @ H @ B ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_954_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > risk_Free_account] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups6033208628184776703ccount @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_955_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_956_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > real] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_957_sum_Omono__neutral__left,axiom,
! [T: set_real,S: set_real,G: real > real] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8097168146408367636l_real @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_958_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > risk_Free_account] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ( groups6033208628184776703ccount @ G @ T )
= ( groups6033208628184776703ccount @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_959_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_960_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > real] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ T )
= ( groups6591440286371151544t_real @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_961_sum_Omono__neutral__right,axiom,
! [T: set_real,S: set_real,G: real > real] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ G @ T )
= ( groups8097168146408367636l_real @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_962_sum_Omono__neutral__cong__left,axiom,
! [T: set_set_nat,S: set_set_nat,H: set_nat > real,G: set_nat > real] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_real ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups5107569545109728110t_real @ G @ S )
= ( groups5107569545109728110t_real @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_963_sum_Omono__neutral__cong__left,axiom,
! [T: set_real,S: set_real,H: real > risk_Free_account,G: real > risk_Free_account] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( H @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ S )
= ( groups8516999891779824987ccount @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_964_sum_Omono__neutral__cong__left,axiom,
! [T: set_set_nat,S: set_set_nat,H: set_nat > risk_Free_account,G: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups5807469391267537845ccount @ G @ S )
= ( groups5807469391267537845ccount @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_965_sum_Omono__neutral__cong__left,axiom,
! [T: set_real,S: set_real,H: real > nat,G: real > nat] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_966_sum_Omono__neutral__cong__left,axiom,
! [T: set_set_nat,S: set_set_nat,H: set_nat > nat,G: set_nat > nat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat @ G @ S )
= ( groups8294997508430121362at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_967_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > risk_Free_account,G: nat > risk_Free_account] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ S )
= ( groups6033208628184776703ccount @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_968_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_969_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > real,G: nat > real] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_real ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_970_sum_Omono__neutral__cong__left,axiom,
! [T: set_real,S: set_real,H: real > real,G: real > real] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_real ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_971_sum_Omono__neutral__cong__right,axiom,
! [T: set_set_nat,S: set_set_nat,G: set_nat > real,H: set_nat > real] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups5107569545109728110t_real @ G @ T )
= ( groups5107569545109728110t_real @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_972_sum_Omono__neutral__cong__right,axiom,
! [T: set_real,S: set_real,G: real > risk_Free_account,H: real > risk_Free_account] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8516999891779824987ccount @ G @ T )
= ( groups8516999891779824987ccount @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_973_sum_Omono__neutral__cong__right,axiom,
! [T: set_set_nat,S: set_set_nat,G: set_nat > risk_Free_account,H: set_nat > risk_Free_account] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups5807469391267537845ccount @ G @ T )
= ( groups5807469391267537845ccount @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_974_sum_Omono__neutral__cong__right,axiom,
! [T: set_real,S: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T )
= ( groups1935376822645274424al_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_975_sum_Omono__neutral__cong__right,axiom,
! [T: set_set_nat,S: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat @ G @ T )
= ( groups8294997508430121362at_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_976_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_z1425366712893667068ccount ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6033208628184776703ccount @ G @ T )
= ( groups6033208628184776703ccount @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_977_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_978_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ T )
= ( groups6591440286371151544t_real @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_979_sum_Omono__neutral__cong__right,axiom,
! [T: set_real,S: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T )
= ( groups8097168146408367636l_real @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_980_sum__mono2,axiom,
! [B: set_set_nat,A: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups5107569545109728110t_real @ F @ A ) @ ( groups5107569545109728110t_real @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_981_sum__mono2,axiom,
! [B: set_real,A: set_real,F: real > nat] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_982_sum__mono2,axiom,
! [B: set_set_nat,A: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F @ A ) @ ( groups8294997508430121362at_nat @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_983_sum__mono2,axiom,
! [B: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_984_sum__mono2,axiom,
! [B: set_nat,A: set_nat,F: nat > real] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_985_sum__mono2,axiom,
! [B: set_real,A: set_real,F: real > real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ F @ B ) ) ) ) ) ).
% sum_mono2
thf(fact_986_pos__le__minus__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_987_pos__minus__divideR__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) @ A3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_988_neg__le__minus__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( real_V1485227260804924795R_real @ C @ A3 ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_989_neg__minus__divideR__le__eq,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) @ A3 )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_990_pos__less__minus__divideR__eq,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B3 ) ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_991_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_992_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_993_ComplI,axiom,
! [C: real,A: set_real] :
( ~ ( member_real @ C @ A )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_994_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_995_ComplI,axiom,
! [C: set_nat,A: set_set_nat] :
( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) ) ) ).
% ComplI
thf(fact_996_Compl__iff,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_997_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_998_Compl__iff,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
= ( ~ ( member_set_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_999_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1000_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A5: set_real] :
( collect_real
@ ^ [X4: real] :
~ ( member_real @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1001_Compl__eq,axiom,
( uminus613421341184616069et_nat
= ( ^ [A5: set_set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
~ ( member_set_nat @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1002_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A5: set_nat] :
( collect_nat
@ ^ [X4: nat] :
~ ( member_nat @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1003_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1004_ComplD,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C @ A ) ) ).
% ComplD
thf(fact_1005_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_1006_ComplD,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
=> ~ ( member_set_nat @ C @ A ) ) ).
% ComplD
thf(fact_1007_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A5: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1008_uminus__set__def,axiom,
( uminus613421341184616069et_nat
= ( ^ [A5: set_set_nat] :
( collect_set_nat
@ ( uminus6401447641752708672_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1009_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A5: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1010_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1011_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1012_minus__set__def,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B5: set_real] :
( collect_real
@ ( minus_minus_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1013_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1014_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1015_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1016_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1017_diff__less__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1018_sum__subtractf__nat,axiom,
! [A: set_real,G: real > nat,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_1019_sum__subtractf__nat,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_1020_sum__subtractf__nat,axiom,
! [A: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A ) @ ( groups8294997508430121362at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_1021_sum__diff__nat,axiom,
! [B: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1022_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_1023_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1024_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1025_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [A2: real,B2: real,C3: real] :
( ( P @ A2 @ B2 )
=> ( ( P @ B2 @ C3 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C3 )
=> ( P @ A2 @ C3 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B3 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A2: real,B2: real] :
( ( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A2 ) @ D3 ) )
=> ( P @ A2 @ B2 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1026_sum_Ozero__middle,axiom,
! [P5: nat,K: nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( ord_less_eq_nat @ one_one_nat @ P5 )
=> ( ( ord_less_eq_nat @ K @ P5 )
=> ( ( groups6033208628184776703ccount
@ ^ [J: nat] : ( if_Risk_Free_account @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_Risk_Free_account @ ( J = K ) @ zero_z1425366712893667068ccount @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P5 ) )
= ( groups6033208628184776703ccount
@ ^ [J: nat] : ( if_Risk_Free_account @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1027_sum_Ozero__middle,axiom,
! [P5: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P5 )
=> ( ( ord_less_eq_nat @ K @ P5 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_nat @ ( J = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P5 ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1028_sum_Ozero__middle,axiom,
! [P5: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P5 )
=> ( ( ord_less_eq_nat @ K @ P5 )
=> ( ( groups6591440286371151544t_real
@ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_real @ ( J = K ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P5 ) )
= ( groups6591440286371151544t_real
@ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1029_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_1030_prod__mono2,axiom,
! [B: set_real,A: set_real,F: real > real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A ) @ ( groups1681761925125756287l_real @ F @ B ) ) ) ) ) ) ).
% prod_mono2
thf(fact_1031_prod__mono2,axiom,
! [B: set_set_nat,A: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups3619160379726066777t_real @ F @ A ) @ ( groups3619160379726066777t_real @ F @ B ) ) ) ) ) ) ).
% prod_mono2
thf(fact_1032_prod__mono2,axiom,
! [B: set_nat,A: set_nat,F: nat > real] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A2 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A ) @ ( groups129246275422532515t_real @ F @ B ) ) ) ) ) ) ).
% prod_mono2
thf(fact_1033_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1034_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1035_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1036_prod__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups708209901874060359at_nat @ F @ A )
= one_one_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_1037_prod__zero__iff,axiom,
! [A: set_nat,F: nat > real] :
( ( finite_finite_nat @ A )
=> ( ( ( groups129246275422532515t_real @ F @ A )
= zero_zero_real )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1038_prod__zero__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups708209901874060359at_nat @ F @ A )
= zero_zero_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1039_prod__pos__nat__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_1040_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1041_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1042_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G @ A )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_1043_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1044_prod_Odelta,axiom,
! [S: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_1045_prod_Odelta,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups4248547760180025341at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups4248547760180025341at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_1046_prod_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_1047_prod_Odelta_H,axiom,
! [S: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S )
=> ( ( ( member_real @ A3 @ S )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_1048_prod_Odelta_H,axiom,
! [S: set_set_nat,A3: set_nat,B3: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A3 @ S )
=> ( ( groups4248547760180025341at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_set_nat @ A3 @ S )
=> ( ( groups4248547760180025341at_nat
@ ^ [K3: set_nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_1049_prod_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( if_nat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_1050_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1051_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1052_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1053_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1054_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1055_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups4696554848551431203al_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1056_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups708209901874060359at_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1057_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > nat,A: set_set_nat] :
( ( ( groups4248547760180025341at_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A2: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( G @ A2 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1058_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1059_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_1060_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1061_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1062_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_1063_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1064_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1065_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1066_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1067_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1068_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1069_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_1070_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1071_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1072_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [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 @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1073_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1074_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1075_prod__mono,axiom,
! [A: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A ) @ ( groups1681761925125756287l_real @ G @ A ) ) ) ).
% prod_mono
thf(fact_1076_prod__mono,axiom,
! [A: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A ) @ ( groups129246275422532515t_real @ G @ A ) ) ) ).
% prod_mono
thf(fact_1077_prod__mono,axiom,
! [A: set_set_nat,F: set_nat > real,G: set_nat > real] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups3619160379726066777t_real @ F @ A ) @ ( groups3619160379726066777t_real @ G @ A ) ) ) ).
% prod_mono
thf(fact_1078_prod__mono,axiom,
! [A: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A ) @ ( groups4696554848551431203al_nat @ G @ A ) ) ) ).
% prod_mono
thf(fact_1079_prod__mono,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A ) @ ( groups708209901874060359at_nat @ G @ A ) ) ) ).
% prod_mono
thf(fact_1080_prod__mono,axiom,
! [A: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4248547760180025341at_nat @ F @ A ) @ ( groups4248547760180025341at_nat @ G @ A ) ) ) ).
% prod_mono
thf(fact_1081_prod__ge__1,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1082_prod__ge__1,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1083_prod__ge__1,axiom,
! [A: set_set_nat,F: set_nat > real] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups3619160379726066777t_real @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1084_prod__ge__1,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1085_prod__ge__1,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1086_prod__ge__1,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4248547760180025341at_nat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_1087_prod__zero,axiom,
! [A: set_nat,F: nat > real] :
( ( finite_finite_nat @ A )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( groups129246275422532515t_real @ F @ A )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_1088_prod__zero,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( F @ X3 )
= zero_zero_nat ) )
=> ( ( groups708209901874060359at_nat @ F @ A )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_1089_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1090_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1091_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1092_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_real @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1093_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1094_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1095_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N5: nat] : ( ord_less_real @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1096_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1097_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_real @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1098_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1099_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1100_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1101_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1102_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1103_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1104_diff__Suc__less,axiom,
! [N2: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1105_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1106_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1107_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1108_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1109_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1110_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J2 )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1111_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J2 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1112_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1113_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1114_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ ( suc @ X2 ) @ A )
=> ( ( F @ ( suc @ X2 ) )
= ( G @ ( suc @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A )
= ( groups6591440286371151544t_real @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_1115_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1116_prod_Ointer__filter,axiom,
! [A: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups4696554848551431203al_nat @ G
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups4696554848551431203al_nat
@ ^ [X4: real] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
@ A ) ) ) ).
% prod.inter_filter
thf(fact_1117_prod_Ointer__filter,axiom,
! [A: set_set_nat,G: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( groups4248547760180025341at_nat @ G
@ ( collect_set_nat
@ ^ [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups4248547760180025341at_nat
@ ^ [X4: set_nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
@ A ) ) ) ).
% prod.inter_filter
thf(fact_1118_prod_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
@ A ) ) ) ).
% prod.inter_filter
thf(fact_1119_prod__le__1,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 @ ( F @ X2 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_1120_prod__le__1,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 @ ( F @ X2 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_1121_prod__le__1,axiom,
! [A: set_set_nat,F: set_nat > real] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) )
& ( ord_less_eq_real @ ( F @ X2 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups3619160379726066777t_real @ F @ A ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_1122_prod__le__1,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 @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1123_prod__le__1,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1124_prod__le__1,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4248547760180025341at_nat @ F @ A ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1125_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S: set_real,I2: real > real,J2: real > real,T: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S )
= ( groups4696554848551431203al_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1126_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_set_nat,S: set_real,I2: set_nat > real,J2: real > set_nat,T: set_set_nat,G: real > nat,H: set_nat > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite1152437895449049373et_nat @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_set_nat @ ( J2 @ A2 ) @ ( minus_2163939370556025621et_nat @ T @ T4 ) ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S )
= ( groups4248547760180025341at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1127_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_set_nat,T4: set_real,S: set_set_nat,I2: real > set_nat,J2: set_nat > real,T: set_real,G: set_nat > nat,H: real > nat] :
( ( finite1152437895449049373et_nat @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_set_nat @ ( I2 @ B2 ) @ ( minus_2163939370556025621et_nat @ S @ S5 ) ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4248547760180025341at_nat @ G @ S )
= ( groups4696554848551431203al_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1128_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_set_nat,T4: set_set_nat,S: set_set_nat,I2: set_nat > set_nat,J2: set_nat > set_nat,T: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ S5 )
=> ( ( finite1152437895449049373et_nat @ T4 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( member_set_nat @ ( J2 @ A2 ) @ ( minus_2163939370556025621et_nat @ T @ T4 ) ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( member_set_nat @ ( I2 @ B2 ) @ ( minus_2163939370556025621et_nat @ S @ S5 ) ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4248547760180025341at_nat @ G @ S )
= ( groups4248547760180025341at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1129_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_nat,S: set_real,I2: nat > real,J2: real > nat,T: set_nat,G: real > nat,H: nat > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S5 ) ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1130_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_set_nat,T4: set_nat,S: set_set_nat,I2: nat > set_nat,J2: set_nat > nat,T: set_nat,G: set_nat > nat,H: nat > nat] :
( ( finite1152437895449049373et_nat @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_set_nat @ ( I2 @ B2 ) @ ( minus_2163939370556025621et_nat @ S @ S5 ) ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups4248547760180025341at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1131_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_real,S: set_nat,I2: real > nat,J2: nat > real,T: set_real,G: nat > nat,H: real > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_real @ ( J2 @ A2 ) @ ( minus_minus_set_real @ T @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups4696554848551431203al_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1132_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_set_nat,S: set_nat,I2: set_nat > nat,J2: nat > set_nat,T: set_set_nat,G: nat > nat,H: set_nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite1152437895449049373et_nat @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_set_nat @ ( J2 @ A2 ) @ ( minus_2163939370556025621et_nat @ T @ T4 ) ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups4248547760180025341at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1133_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T4: set_nat,S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ S @ S5 ) )
=> ( member_nat @ ( J2 @ A2 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I2 @ B2 ) @ ( minus_minus_set_nat @ S @ S5 ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S5 )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T4 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1134_prod_Ozero__middle,axiom,
! [P5: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P5 )
=> ( ( ord_less_eq_nat @ K @ P5 )
=> ( ( groups708209901874060359at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_nat @ ( J = K ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P5 ) )
= ( groups708209901874060359at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_1135_prod_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= one_one_nat ) ) ) )
= ( groups708209901874060359at_nat @ G @ A ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_1136_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N2 )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1137_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1138_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1139_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1140_sum__eq__Suc0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1141_less__1__prod2,axiom,
! [I4: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I4 )
=> ( ( member_real @ I2 @ I4 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_1142_less__1__prod2,axiom,
! [I4: set_set_nat,I2: set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups3619160379726066777t_real @ F @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_1143_less__1__prod2,axiom,
! [I4: set_nat,I2: nat,F: nat > real] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_1144_prod_Osame__carrier,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ A )
= ( groups4696554848551431203al_nat @ H @ B ) )
= ( ( groups4696554848551431203al_nat @ G @ C2 )
= ( groups4696554848551431203al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1145_prod_Osame__carrier,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4248547760180025341at_nat @ G @ A )
= ( groups4248547760180025341at_nat @ H @ B ) )
= ( ( groups4248547760180025341at_nat @ G @ C2 )
= ( groups4248547760180025341at_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1146_prod_Osame__carrier,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G @ A )
= ( groups708209901874060359at_nat @ H @ B ) )
= ( ( groups708209901874060359at_nat @ G @ C2 )
= ( groups708209901874060359at_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1147_prod_Osame__carrierI,axiom,
! [C2: set_real,A: set_real,B: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [A2: real] :
( ( member_real @ A2 @ ( minus_minus_set_real @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ C2 )
= ( groups4696554848551431203al_nat @ H @ C2 ) )
=> ( ( groups4696554848551431203al_nat @ G @ A )
= ( groups4696554848551431203al_nat @ H @ B ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1148_prod_Osame__carrierI,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [A2: set_nat] :
( ( member_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups4248547760180025341at_nat @ G @ C2 )
= ( groups4248547760180025341at_nat @ H @ C2 ) )
=> ( ( groups4248547760180025341at_nat @ G @ A )
= ( groups4248547760180025341at_nat @ H @ B ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1149_prod_Osame__carrierI,axiom,
! [C2: set_nat,A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G @ A2 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C2 @ B ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G @ C2 )
= ( groups708209901874060359at_nat @ H @ C2 ) )
=> ( ( groups708209901874060359at_nat @ G @ A )
= ( groups708209901874060359at_nat @ H @ B ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1150_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ G @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_1151_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ T )
= ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_1152_prod_Omono__neutral__cong__left,axiom,
! [T: set_real,S: set_real,H: real > nat,G: real > nat] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( H @ X2 )
= one_one_nat ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S )
= ( groups4696554848551431203al_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1153_prod_Omono__neutral__cong__left,axiom,
! [T: set_set_nat,S: set_set_nat,H: set_nat > nat,G: set_nat > nat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H @ X2 )
= one_one_nat ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4248547760180025341at_nat @ G @ S )
= ( groups4248547760180025341at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1154_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1155_prod_Omono__neutral__cong__right,axiom,
! [T: set_real,S: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ T )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( minus_minus_set_real @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ T )
= ( groups4696554848551431203al_nat @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1156_prod_Omono__neutral__cong__right,axiom,
! [T: set_set_nat,S: set_set_nat,G: set_nat > nat,H: set_nat > nat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4248547760180025341at_nat @ G @ T )
= ( groups4248547760180025341at_nat @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1157_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ T )
= ( groups708209901874060359at_nat @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1158_sum__telescope,axiom,
! [F: nat > risk_Free_account,I2: nat] :
( ( groups6033208628184776703ccount
@ ^ [I: nat] : ( minus_4846202936726426316ccount @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_4846202936726426316ccount @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_1159_sum__telescope,axiom,
! [F: nat > real,I2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_1160_norm__prod__diff,axiom,
! [I4: set_set_nat,Z2: set_nat > real,W3: set_nat > real] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups3619160379726066777t_real @ Z2 @ I4 ) @ ( groups3619160379726066777t_real @ W3 @ I4 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I: set_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1161_norm__prod__diff,axiom,
! [I4: set_set_nat,Z2: set_nat > complex,W3: set_nat > complex] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups1092910753850256091omplex @ Z2 @ I4 ) @ ( groups1092910753850256091omplex @ W3 @ I4 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I: set_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1162_norm__prod__diff,axiom,
! [I4: set_nat,Z2: nat > real,W3: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z2 @ I4 ) @ ( groups129246275422532515t_real @ W3 @ I4 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1163_norm__prod__diff,axiom,
! [I4: set_nat,Z2: nat > complex,W3: nat > complex] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z2 @ I4 ) @ ( groups6464643781859351333omplex @ W3 @ I4 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1164_norm__prod__diff,axiom,
! [I4: set_real,Z2: real > real,W3: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z2 @ I4 ) @ ( groups1681761925125756287l_real @ W3 @ I4 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1165_norm__prod__diff,axiom,
! [I4: set_real,Z2: real > complex,W3: real > complex] :
( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W3 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z2 @ I4 ) @ ( groups713298508707869441omplex @ W3 @ I4 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I ) @ ( W3 @ I ) ) )
@ I4 ) ) ) ) ).
% norm_prod_diff
thf(fact_1166_sum__telescope_H_H,axiom,
! [M2: nat,N2: nat,F: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( minus_4846202936726426316ccount @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
= ( minus_4846202936726426316ccount @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_1167_sum__telescope_H_H,axiom,
! [M2: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
= ( minus_minus_real @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_1168_Cauchy__iff,axiom,
( topolo4055970368930404560y_real
= ( ^ [X7: nat > real] :
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M6: nat] :
! [M: nat] :
( ( ord_less_eq_nat @ M6 @ M )
=> ! [N: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X7 @ M ) @ ( X7 @ N ) ) ) @ E2 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_1169_Cauchy__iff,axiom,
( topolo6517432010174082258omplex
= ( ^ [X7: nat > complex] :
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M6: nat] :
! [M: nat] :
( ( ord_less_eq_nat @ M6 @ M )
=> ! [N: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X7 @ M ) @ ( X7 @ N ) ) ) @ E2 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_1170_CauchyI,axiom,
! [X5: nat > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M7 @ N5 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X5 @ M5 ) @ ( X5 @ N5 ) ) ) @ E ) ) ) )
=> ( topolo4055970368930404560y_real @ X5 ) ) ).
% CauchyI
thf(fact_1171_CauchyI,axiom,
! [X5: nat > complex] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M7 @ N5 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X5 @ M5 ) @ ( X5 @ N5 ) ) ) @ E ) ) ) )
=> ( topolo6517432010174082258omplex @ X5 ) ) ).
% CauchyI
thf(fact_1172_CauchyD,axiom,
! [X5: nat > real,E3: real] :
( ( topolo4055970368930404560y_real @ X5 )
=> ( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ M8 @ M3 )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ M8 @ N7 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X5 @ M3 ) @ ( X5 @ N7 ) ) ) @ E3 ) ) ) ) ) ).
% CauchyD
thf(fact_1173_CauchyD,axiom,
! [X5: nat > complex,E3: real] :
( ( topolo6517432010174082258omplex @ X5 )
=> ( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ M8 @ M3 )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ M8 @ N7 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X5 @ M3 ) @ ( X5 @ N7 ) ) ) @ E3 ) ) ) ) ) ).
% CauchyD
thf(fact_1174_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U2: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_1175_atLeastAtMost__iff,axiom,
! [I2: set_nat,L: set_nat,U2: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U2 ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_1176_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U2: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_1177_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1178_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1179_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1180_finite__atLeastAtMost,axiom,
! [L: nat,U2: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U2 ) ) ).
% finite_atLeastAtMost
thf(fact_1181_atLeastatMost__subset__iff,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( ( ord_less_eq_real @ C @ A3 )
& ( ord_less_eq_real @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1182_atLeastatMost__subset__iff,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A3 @ B3 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A3 @ B3 )
| ( ( ord_less_eq_set_nat @ C @ A3 )
& ( ord_less_eq_set_nat @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1183_atLeastatMost__subset__iff,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( ( ord_less_eq_nat @ C @ A3 )
& ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1184_infinite__Icc__iff,axiom,
! [A3: real,B3: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% infinite_Icc_iff
thf(fact_1185_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1186_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1187_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1188_not__Iic__eq__Icc,axiom,
! [H2: real,L: real,H: real] :
( ( set_ord_atMost_real @ H2 )
!= ( set_or1222579329274155063t_real @ L @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_1189_infinite__Icc,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) ) ).
% infinite_Icc
thf(fact_1190_not__Iic__le__Icc,axiom,
! [H: real,L2: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_1191_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1192_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1193_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_1194_atLeastatMost__psubset__iff,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( ( ord_less_eq_real @ C @ A3 )
& ( ord_less_eq_real @ B3 @ D )
& ( ( ord_less_real @ C @ A3 )
| ( ord_less_real @ B3 @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1195_atLeastatMost__psubset__iff,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A3 @ B3 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A3 @ B3 )
| ( ( ord_less_eq_set_nat @ C @ A3 )
& ( ord_less_eq_set_nat @ B3 @ D )
& ( ( ord_less_set_nat @ C @ A3 )
| ( ord_less_set_nat @ B3 @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1196_atLeastatMost__psubset__iff,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( ( ord_less_eq_nat @ C @ A3 )
& ( ord_less_eq_nat @ B3 @ D )
& ( ( ord_less_nat @ C @ A3 )
| ( ord_less_nat @ B3 @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1197_sum__shift__lb__Suc0__0,axiom,
! [F: nat > risk_Free_account,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_z1425366712893667068ccount )
=> ( ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1198_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1199_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1200_sum__Suc__diff,axiom,
! [M2: nat,N2: nat,F: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ( groups6033208628184776703ccount
@ ^ [I: nat] : ( minus_4846202936726426316ccount @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= ( minus_4846202936726426316ccount @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_1201_sum__Suc__diff,axiom,
! [M2: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_1202_sum__natinterval__diff,axiom,
! [M2: nat,N2: nat,F: nat > risk_Free_account] :
( ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( minus_4846202936726426316ccount @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= ( minus_4846202936726426316ccount @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( minus_4846202936726426316ccount @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= zero_z1425366712893667068ccount ) ) ) ).
% sum_natinterval_diff
thf(fact_1203_sum__natinterval__diff,axiom,
! [M2: nat,N2: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_1204_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1205_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1206_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1207_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > risk_Free_account] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= zero_z1425366712893667068ccount ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1208_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1209_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1210_add__left__cancel,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A3 @ B3 )
= ( plus_p1863581527469039996ccount @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1211_add__left__cancel,axiom,
! [A3: real,B3: real,C: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1212_add__left__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1213_add__left__cancel,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= ( plus_plus_complex @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_1214_add__right__cancel,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= ( plus_plus_complex @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_1215_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1216_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1217_Rep__account__plus,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) )
= ( ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N ) ) ) ) ).
% Rep_account_plus
thf(fact_1218_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1219_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1220_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1221_real__add__minus__iff,axiom,
! [X: real,A3: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A3 ) )
= zero_zero_real )
= ( X = A3 ) ) ).
% real_add_minus_iff
thf(fact_1222_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1223_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1224_additive__strictly__solvent,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_12 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_22 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_1225_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1226_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1227_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_1228_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1229_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1230_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1231_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1232_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1233_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1234_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1235_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N ) ) ) ) ) ).
% plus_account_def
thf(fact_1236_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1237_minus__account__def,axiom,
( minus_4846202936726426316ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] : ( plus_p1863581527469039996ccount @ Alpha_1 @ ( uminus3377898441596595772ccount @ Alpha_2 ) ) ) ) ).
% minus_account_def
thf(fact_1238_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1239_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1240_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1241_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1242_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1243_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K4: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1244_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1245_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1246_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1247_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1248_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1249_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1250_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1251_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1252_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1253_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1254_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1255_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X4: real,Y4: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y4 ) ) ) ) ).
% minus_real_def
thf(fact_1256_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1257_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1258_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1259_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1260_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1261_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1262_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1263_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1264_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ( ( ord_less_nat @ A3 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1265_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1266_complex__mod__triangle__ineq2,axiom,
! [B3: complex,A3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B3 @ A3 ) ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ A3 ) ) ).
% complex_mod_triangle_ineq2
% 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] :
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ x ) @ ( set_ord_atMost_nat @ N5 ) )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ y ) @ ( set_ord_atMost_nat @ N5 ) ) ) ).
%------------------------------------------------------------------------------