TPTP Problem File: SLH0149^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Risk_Free_Lending/0000_Risk_Free_Lending/prob_00212_006881__5762416_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1363 ( 614 unt; 88 typ; 0 def)
% Number of atoms : 3511 (1237 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10726 ( 280 ~; 86 |; 203 &;8751 @)
% ( 0 <=>;1406 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 521 ( 521 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 82 usr; 11 con; 0-4 aty)
% Number of variables : 3542 ( 280 ^;3206 !; 56 ?;3542 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:56:35.341
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (82)
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_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_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_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__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_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_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__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__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__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__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_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_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_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_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_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Risk____Free____Lending__Oaccount,type,
set_fo6648016647582781957ccount: ( nat > risk_Free_account > risk_Free_account ) > nat > nat > risk_Free_account > risk_Free_account ).
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_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
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_m____,type,
m: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_x____,type,
x: risk_Free_account ).
thf(sy_v_y____,type,
y: risk_Free_account ).
% Relevant facts (1267)
thf(fact_0__C_092_060star_062_C,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ x ) @ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ y ) @ ( set_ord_atMost_nat @ N ) ) ) ).
% "\<star>"
thf(fact_1_False,axiom,
n != zero_zero_nat ).
% False
thf(fact_2__092_060open_062x_A_092_060le_062_Ay_092_060close_062,axiom,
ord_le4245800335709223507ccount @ x @ y ).
% \<open>x \<le> y\<close>
thf(fact_3__092_060open_062y_A_092_060le_062_Ax_092_060close_062,axiom,
ord_le4245800335709223507ccount @ y @ x ).
% \<open>y \<le> x\<close>
thf(fact_4_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_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_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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24__092_060open_062n_A_061_Am_A_L_A1_092_060close_062,axiom,
( n
= ( plus_plus_nat @ m @ one_one_nat ) ) ).
% \<open>n = m + 1\<close>
thf(fact_25_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_26__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_An_A_061_Am_A_L_A1_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: nat] :
( n
!= ( plus_plus_nat @ M @ one_one_nat ) ) ).
% \<open>\<And>thesis. (\<And>m. n = m + 1 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_27_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N2 ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% less_eq_account_def
thf(fact_28_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_29_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_30_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_31_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_32_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_33_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_34_scaleR__zero__right,axiom,
! [A3: real] :
( ( real_V1485227260804924795R_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_43_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_44_of__real__1,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_1
thf(fact_45_of__real__eq__1__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_46_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_add
thf(fact_47_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_48_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [Uu: real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_49_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_of__real__def,axiom,
( real_V1803761363581548252l_real
= ( ^ [R: real] : ( real_V1485227260804924795R_real @ R @ one_one_real ) ) ) ).
% of_real_def
thf(fact_62_scaleR__left__distrib,axiom,
! [A3: real,B3: real,X: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A3 @ B3 ) @ X )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ B3 @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_63_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_64_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_65_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_66_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_67_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_68_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_69_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_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_74_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_75_scaleR__right__distrib,axiom,
! [A3: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A3 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ ( real_V1485227260804924795R_real @ A3 @ Y ) ) ) ).
% scaleR_right_distrib
thf(fact_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A ) @ ( groups6591440286371151544t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_90_sum_Odistrib,axiom,
! [G: real > real,H: real > real,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [X4: real] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_real @ ( groups8097168146408367636l_real @ G @ A ) @ ( groups8097168146408367636l_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_91_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_92_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_93_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_94_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_95_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_96_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_97_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_98_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_99_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_100_add__le__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_101_add__le__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_102_add__le__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_103_add__le__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_104_le__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_105_le__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_106_le__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_107_le__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_108_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_109_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_110_add__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_111_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_112_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_113_add_Oright__neutral,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% add.right_neutral
thf(fact_114_add_Oright__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.right_neutral
thf(fact_115_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_116_add__cancel__left__left,axiom,
! [B3: nat,A3: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_117_add__cancel__left__left,axiom,
! [B3: risk_Free_account,A3: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B3 @ A3 )
= A3 )
= ( B3 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_left
thf(fact_118_add__cancel__left__left,axiom,
! [B3: real,A3: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_119_add__cancel__left__right,axiom,
! [A3: nat,B3: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_120_add__cancel__left__right,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A3 @ B3 )
= A3 )
= ( B3 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_right
thf(fact_121_add__cancel__left__right,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_122_add__right__cancel,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_123_add__right__cancel,axiom,
! [B3: risk_Free_account,A3: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B3 @ A3 )
= ( plus_p1863581527469039996ccount @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_124_add__right__cancel,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_125_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_126_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_127_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_128_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_129_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_130_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_131_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_132_add__le__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_133_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_134_add__le__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_135_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_136_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_137_add__0,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A3 )
= A3 ) ).
% add_0
thf(fact_138_add__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add_0
thf(fact_139_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_140_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_141_add__cancel__right__right,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_142_add__cancel__right__right,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( A3
= ( plus_p1863581527469039996ccount @ A3 @ B3 ) )
= ( B3 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_right
thf(fact_143_add__cancel__right__right,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ A3 @ B3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_144_add__cancel__right__left,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_145_add__cancel__right__left,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( A3
= ( plus_p1863581527469039996ccount @ B3 @ A3 ) )
= ( B3 = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_left
thf(fact_146_add__cancel__right__left,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ B3 @ A3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_147_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_148_Rep__account__plus,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) )
= ( ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N2 ) ) ) ) ).
% Rep_account_plus
thf(fact_149_scaleR__eq__iff,axiom,
! [B3: real,U2: real,A3: real] :
( ( ( plus_plus_real @ B3 @ ( real_V1485227260804924795R_real @ U2 @ A3 ) )
= ( plus_plus_real @ A3 @ ( real_V1485227260804924795R_real @ U2 @ B3 ) ) )
= ( ( A3 = B3 )
| ( U2 = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_150_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_151_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_152_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_153_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_154_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_155_trans__le__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_156_trans__le__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_157_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_158_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_159_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_160_add__leD2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ K @ N3 ) ) ).
% add_leD2
thf(fact_161_add__leD1,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% add_leD1
thf(fact_162_le__add2,axiom,
! [N3: nat,M2: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M2 @ N3 ) ) ).
% le_add2
thf(fact_163_le__add1,axiom,
! [N3: nat,M2: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M2 ) ) ).
% le_add1
thf(fact_164_add__leE,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N3 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N3 )
=> ~ ( ord_less_eq_nat @ K @ N3 ) ) ) ).
% add_leE
thf(fact_165_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_166_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N2 ) ) ) ) ) ).
% plus_account_def
thf(fact_167_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_168_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_169_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_170_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_171_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_172_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_173_add__right__imp__eq,axiom,
! [B3: risk_Free_account,A3: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B3 @ A3 )
= ( plus_p1863581527469039996ccount @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_174_add__right__imp__eq,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_175_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_176_add__left__imp__eq,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_imp_eq
thf(fact_177_add__left__imp__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_178_add_Oleft__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_179_add_Oleft__commute,axiom,
! [B3: risk_Free_account,A3: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ B3 @ ( plus_p1863581527469039996ccount @ A3 @ C ) )
= ( plus_p1863581527469039996ccount @ A3 @ ( plus_p1863581527469039996ccount @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_180_add_Oleft__commute,axiom,
! [B3: real,A3: real,C: real] :
( ( plus_plus_real @ B3 @ ( plus_plus_real @ A3 @ C ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_181_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_182_add_Ocommute,axiom,
( plus_p1863581527469039996ccount
= ( ^ [A4: risk_Free_account,B4: risk_Free_account] : ( plus_p1863581527469039996ccount @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_183_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_184_add_Oright__cancel,axiom,
! [B3: risk_Free_account,A3: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B3 @ A3 )
= ( plus_p1863581527469039996ccount @ C @ A3 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_185_add_Oright__cancel,axiom,
! [B3: real,A3: real,C: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C @ A3 ) )
= ( B3 = C ) ) ).
% add.right_cancel
thf(fact_186_add_Oleft__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_187_add_Oleft__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_188_add_Oassoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_189_add_Oassoc,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ C )
= ( plus_p1863581527469039996ccount @ A3 @ ( plus_p1863581527469039996ccount @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_190_add_Oassoc,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_191_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B3: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_192_group__cancel_Oadd2,axiom,
! [B: risk_Free_account,K: risk_Free_account,B3: risk_Free_account,A3: risk_Free_account] :
( ( B
= ( plus_p1863581527469039996ccount @ K @ B3 ) )
=> ( ( plus_p1863581527469039996ccount @ A3 @ B )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_193_group__cancel_Oadd2,axiom,
! [B: real,K: real,B3: real,A3: real] :
( ( B
= ( plus_plus_real @ K @ B3 ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_194_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_195_group__cancel_Oadd1,axiom,
! [A: risk_Free_account,K: risk_Free_account,A3: risk_Free_account,B3: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ K @ A3 ) )
=> ( ( plus_p1863581527469039996ccount @ A @ B3 )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_196_group__cancel_Oadd1,axiom,
! [A: real,K: real,A3: real,B3: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_197_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_198_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_200_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ C )
= ( plus_p1863581527469039996ccount @ A3 @ ( plus_p1863581527469039996ccount @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_201_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_202_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_203_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_204_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_205_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_206_add__le__imp__le__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_207_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_208_add__le__imp__le__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_209_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_210_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C2: nat] :
( B4
= ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_211_add__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_212_add__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_213_less__eqE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( B3
!= ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% less_eqE
thf(fact_214_add__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_215_add__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_216_add__mono,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 @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_217_add__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_mono
thf(fact_218_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_219_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_220_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_221_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_222_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_223_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_224_add_Ogroup__left__neutral,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_225_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_226_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_227_add_Ocomm__neutral,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% add.comm_neutral
thf(fact_228_add_Ocomm__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.comm_neutral
thf(fact_229_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_230_comm__monoid__add__class_Oadd__0,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_231_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_232_add__eq__self__zero,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= M2 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_233_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_234_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_235_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_236_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_237_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_238_add__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 @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_239_add__nonpos__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_240_add__nonneg__nonneg,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 @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_241_add__nonneg__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_242_add__increasing2,axiom,
! [C: real,B3: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_243_add__increasing2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_244_add__decreasing2,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_245_add__decreasing2,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_246_add__increasing,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_247_add__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_248_add__decreasing,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_249_add__decreasing,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_250_double__eq__0__iff,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_251_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_252_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_253_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_262_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_263_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_264_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_265_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_266_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_267_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_268_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_269_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_270_eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( M2 = N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% eq_imp_le
thf(fact_271_le__antisym,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M2 )
=> ( M2 = N3 ) ) ) ).
% le_antisym
thf(fact_272_nat__le__linear,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
| ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% nat_le_linear
thf(fact_273_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_274_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_275_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_276_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_277_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_278_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_279_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_280_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_281_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_282_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_296_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_297_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_298_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_299_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_300_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_301_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_302_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [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_303_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_304_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_305_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_306_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_307_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_308_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_309_antisym,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_310_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_311_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_312_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_313_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_314_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_334_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_335_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_336_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_337_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [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_363_subset__trans,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_eq_set_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_364_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_365_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_366_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_367_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_368_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_369_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_370_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_371_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_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_382_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_383_add__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_384_add__0__iff,axiom,
! [B3: real,A3: real] :
( ( B3
= ( plus_plus_real @ B3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_385_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_386_verit__sum__simplify,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% verit_sum_simplify
thf(fact_387_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_388_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_389_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_390_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_391_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_392_sum_Oub__add__nat,axiom,
! [M2: nat,N3: nat,G: nat > nat,P2: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N3 @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N3 @ P2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_393_sum_Oub__add__nat,axiom,
! [M2: nat,N3: nat,G: nat > risk_Free_account,P2: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N3 @ one_one_nat ) )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N3 @ P2 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_394_sum_Oub__add__nat,axiom,
! [M2: nat,N3: nat,G: nat > real,P2: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N3 @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N3 @ P2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_395_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_411_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M2: nat,K: nat,N3: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_412_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_413_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_414_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_415_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_416_verit__comp__simplify1_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_417_is__num__normalize_I1_J,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_418_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_419_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_420_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_421_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_422_ex__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( P @ M3 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_423_all__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( P @ M3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_424_pred__subset__eq,axiom,
! [R2: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ R2 )
@ ^ [X4: real] : ( member_real @ X4 @ S ) )
= ( ord_less_eq_set_real @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_425_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_426_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_427_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_428_sum__atLeastAtMost__code,axiom,
! [F: nat > risk_Free_account,A3: nat,B3: nat] :
( ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo6648016647582781957ccount
@ ^ [A4: nat] : ( plus_p1863581527469039996ccount @ ( F @ A4 ) )
@ A3
@ B3
@ zero_z1425366712893667068ccount ) ) ).
% sum_atLeastAtMost_code
thf(fact_429_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A3: nat,B3: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_430_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_431_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_432_sum__natinterval__diff,axiom,
! [M2: nat,N3: nat,F: nat > risk_Free_account] :
( ( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( minus_4846202936726426316ccount @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( minus_4846202936726426316ccount @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6033208628184776703ccount
@ ^ [K3: nat] : ( minus_4846202936726426316ccount @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= zero_z1425366712893667068ccount ) ) ) ).
% sum_natinterval_diff
thf(fact_433_sum__natinterval__diff,axiom,
! [M2: nat,N3: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_434_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_435_sum__up__index__split,axiom,
! [F: nat > nat,M2: nat,N3: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N3 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_436_sum__up__index__split,axiom,
! [F: nat > risk_Free_account,M2: nat,N3: nat] :
( ( groups6033208628184776703ccount @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6033208628184776703ccount @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N3 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_437_sum__up__index__split,axiom,
! [F: nat > real,M2: nat,N3: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N3 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_438_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_439_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_440_prop__restrict,axiom,
! [X: real,Z3: set_real,X5: set_real,P: real > $o] :
( ( member_real @ X @ Z3 )
=> ( ( ord_less_eq_set_real @ Z3
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_441_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_442_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_443_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_444_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_445_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_446_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_447_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_448_diff__Suc__Suc,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_449_Suc__diff__diff,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_450_diff__diff__cancel,axiom,
! [I2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_451_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_452_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_453_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_454_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_455_diff__zero,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% diff_zero
thf(fact_456_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_457_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_458_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_459_diff__0__right,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ zero_z1425366712893667068ccount )
= A3 ) ).
% diff_0_right
thf(fact_460_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_461_diff__self,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ A3 )
= zero_z1425366712893667068ccount ) ).
% diff_self
thf(fact_462_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_463_add__diff__cancel__right_H,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_464_add__diff__cancel__right_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_465_add__diff__cancel__right_H,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_466_add__diff__cancel__right,axiom,
! [A3: risk_Free_account,C: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ C ) @ ( plus_p1863581527469039996ccount @ B3 @ C ) )
= ( minus_4846202936726426316ccount @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_467_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_468_add__diff__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( minus_minus_real @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_469_add__diff__cancel__left_H,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_470_add__diff__cancel__left_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_471_add__diff__cancel__left_H,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_472_add__diff__cancel__left,axiom,
! [C: risk_Free_account,A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ C @ A3 ) @ ( plus_p1863581527469039996ccount @ C @ B3 ) )
= ( minus_4846202936726426316ccount @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_473_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_474_add__diff__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( minus_minus_real @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_475_diff__add__cancel,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) @ B3 )
= A3 ) ).
% diff_add_cancel
thf(fact_476_diff__add__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% diff_add_cancel
thf(fact_477_add__diff__cancel,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel
thf(fact_478_add__diff__cancel,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel
thf(fact_479_diff__is__0__eq,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_480_diff__is__0__eq_H,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_481_Suc__le__mono,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% Suc_le_mono
thf(fact_482_add__Suc__right,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_483_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_484_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_485_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_486_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_487_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_488_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_489_le__add__diff__inverse2,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_490_le__add__diff__inverse2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_491_le__add__diff__inverse,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_492_le__add__diff__inverse,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_493_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_494_diff__add__zero,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_495_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_496_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_497_scaleR__collapse,axiom,
! [U2: real,A3: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U2 ) @ A3 ) @ ( real_V1485227260804924795R_real @ U2 @ A3 ) )
= A3 ) ).
% scaleR_collapse
thf(fact_498_sum_OatMost__Suc,axiom,
! [G: nat > nat,N3: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_499_sum_OatMost__Suc,axiom,
! [G: nat > risk_Free_account,N3: nat] :
( ( groups6033208628184776703ccount @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_500_sum_OatMost__Suc,axiom,
! [G: nat > real,N3: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_501_scaleR__left_Odiff,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_502_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_503_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_504_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_505_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_506_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_507_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_508_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_509_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_510_Suc__diff__le,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_eq_nat @ N3 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_511_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_512_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_513_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_514_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_515_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: risk_Free_account,Z2: risk_Free_account] : ( Y4 = Z2 ) )
= ( ^ [A4: risk_Free_account,B4: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A4 @ B4 )
= zero_z1425366712893667068ccount ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_516_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_517_diff__diff__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) @ C )
= ( minus_4846202936726426316ccount @ A3 @ ( plus_p1863581527469039996ccount @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_518_diff__diff__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_519_diff__diff__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( minus_minus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% diff_diff_eq
thf(fact_520_add__implies__diff,axiom,
! [C: risk_Free_account,B3: risk_Free_account,A3: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ C @ B3 )
= A3 )
=> ( C
= ( minus_4846202936726426316ccount @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_521_add__implies__diff,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_522_add__implies__diff,axiom,
! [C: real,B3: real,A3: real] :
( ( ( plus_plus_real @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_real @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_523_diff__add__eq__diff__diff__swap,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ ( plus_p1863581527469039996ccount @ B3 @ C ) )
= ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_524_diff__add__eq__diff__diff__swap,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_525_diff__add__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) @ C )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_526_diff__add__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ).
% diff_add_eq
thf(fact_527_diff__diff__eq2,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ ( minus_4846202936726426316ccount @ B3 @ C ) )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_528_diff__diff__eq2,axiom,
! [A3: real,B3: real,C: real] :
( ( minus_minus_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ).
% diff_diff_eq2
thf(fact_529_add__diff__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ ( minus_4846202936726426316ccount @ B3 @ C ) )
= ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_530_add__diff__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( plus_plus_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% add_diff_eq
thf(fact_531_eq__diff__eq,axiom,
! [A3: risk_Free_account,C: risk_Free_account,B3: risk_Free_account] :
( ( A3
= ( minus_4846202936726426316ccount @ C @ B3 ) )
= ( ( plus_p1863581527469039996ccount @ A3 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_532_eq__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( A3
= ( minus_minus_real @ C @ B3 ) )
= ( ( plus_plus_real @ A3 @ B3 )
= C ) ) ).
% eq_diff_eq
thf(fact_533_diff__eq__eq,axiom,
! [A3: risk_Free_account,B3: risk_Free_account,C: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A3 @ B3 )
= C )
= ( A3
= ( plus_p1863581527469039996ccount @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_534_diff__eq__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= C )
= ( A3
= ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_eq_eq
thf(fact_535_group__cancel_Osub1,axiom,
! [A: risk_Free_account,K: risk_Free_account,A3: risk_Free_account,B3: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ K @ A3 ) )
=> ( ( minus_4846202936726426316ccount @ A @ B3 )
= ( plus_p1863581527469039996ccount @ K @ ( minus_4846202936726426316ccount @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_536_group__cancel_Osub1,axiom,
! [A: real,K: real,A3: real,B3: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( minus_minus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.sub1
thf(fact_537_Diff__mono,axiom,
! [A: set_nat,C4: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_538_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_539_double__diff,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_540_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M: nat] :
( N3
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_541_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_542_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_543_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_544_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_545_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N3: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_546_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_547_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_548_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_549_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_550_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_551_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_552_diffs0__imp__equal,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M2 )
= zero_zero_nat )
=> ( M2 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_553_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_554_transitive__stepwise__le,axiom,
! [M2: nat,N3: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z4: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z4 )
=> ( R2 @ X2 @ Z4 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M2 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_555_nat__induct__at__least,axiom,
! [M2: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_556_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_557_not__less__eq__eq,axiom,
! [M2: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_558_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_559_le__Suc__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M2 @ N3 )
| ( M2
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_560_Suc__le__D,axiom,
! [N3: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M6 )
=> ? [M: nat] :
( M6
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_561_le__SucI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_562_le__SucE,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N3 )
=> ( M2
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_563_Suc__leD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% Suc_leD
thf(fact_564_diff__le__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_565_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_566_diff__le__self,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ).
% diff_le_self
thf(fact_567_diff__le__mono,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_568_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_569_le__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_570_eq__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M2 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_571_add__Suc__shift,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N3 )
= ( plus_plus_nat @ M2 @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_572_add__Suc,axiom,
! [M2: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N3 )
= ( suc @ ( plus_plus_nat @ M2 @ N3 ) ) ) ).
% add_Suc
thf(fact_573_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_574_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_575_diff__add__inverse2,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N3 ) @ N3 )
= M2 ) ).
% diff_add_inverse2
thf(fact_576_diff__add__inverse,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ N3 )
= M2 ) ).
% diff_add_inverse
thf(fact_577_diff__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_cancel2
thf(fact_578_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_579_sum__telescope_H_H,axiom,
! [M2: nat,N3: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N3 ) )
= ( minus_minus_real @ ( F @ N3 ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_580_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_581_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_582_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_583_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_584_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_585_sum__Suc__diff,axiom,
! [M2: nat,N3: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_586_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_587_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_588_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A3 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_589_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B3 @ A3 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_590_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_591_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_592_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_593_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_594_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_595_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_596_le__add__diff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_597_diff__add,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% diff_add
thf(fact_598_le__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% le_diff_eq
thf(fact_599_diff__le__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( ord_less_eq_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_le_eq
thf(fact_600_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N3: real,J2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J2 @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J2 @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N3 @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_601_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N3: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_602_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N3 )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N3 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_603_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N3 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_604_lift__Suc__antimono__le,axiom,
! [F: nat > real,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_605_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_606_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_607_lift__Suc__mono__le,axiom,
! [F: nat > real,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_608_lift__Suc__mono__le,axiom,
! [F: nat > nat,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_609_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_610_one__is__add,axiom,
! [M2: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N3 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_611_add__is__1,axiom,
! [M2: nat,N3: nat] :
( ( ( plus_plus_nat @ M2 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_612_diff__add__0,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_613_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_614_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_615_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_616_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_617_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_618_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_619_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_620_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_621_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_622_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_623_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N3: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N3 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_624_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A3: real,E: real,C: real,B3: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A3 @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B3 @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B3 @ A3 ) @ E ) @ D ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_625_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A3: real,E: real,C: real,B3: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A3 @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B3 @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A3 @ B3 ) @ E ) @ C ) @ D ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_626_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N3: nat,M2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N3 @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N3 ) @ I ) )
@ ( set_or1269000886237332187st_nat @ N3 @ M2 ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_627_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_628_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_629_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_630_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_631_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N3: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_632_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > risk_Free_account,N3: nat] :
( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_633_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N3: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_634_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N3: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N3 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_635_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N3: nat,G: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( plus_p1863581527469039996ccount @ ( G @ M2 ) @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N3 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_636_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N3: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) )
= ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N3 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_637_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N3: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_638_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N3: nat,G: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( G @ ( suc @ N3 ) ) @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_639_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N3: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N3 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_640_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N3: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( G @ M2 )
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_641_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N3: nat,G: nat > risk_Free_account] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( G @ M2 )
@ ( groups6033208628184776703ccount
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_642_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N3: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( G @ M2 )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_643_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N3: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N3 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_644_sum_OatMost__Suc__shift,axiom,
! [G: nat > risk_Free_account,N3: nat] :
( ( groups6033208628184776703ccount @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( G @ zero_zero_nat )
@ ( groups6033208628184776703ccount
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N3 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_645_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N3: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N3 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_646_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( 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 @ P2 ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_647_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > risk_Free_account,H: nat > risk_Free_account] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( 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 @ P2 ) )
= ( groups6033208628184776703ccount
@ ^ [J: nat] : ( if_Risk_Free_account @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_648_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( 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 @ P2 ) )
= ( groups6591440286371151544t_real
@ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_649_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_650_add__diff__add,axiom,
! [A3: risk_Free_account,C: risk_Free_account,B3: risk_Free_account,D: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( plus_p1863581527469039996ccount @ A3 @ C ) @ ( plus_p1863581527469039996ccount @ B3 @ D ) )
= ( plus_p1863581527469039996ccount @ ( minus_4846202936726426316ccount @ A3 @ B3 ) @ ( minus_4846202936726426316ccount @ C @ D ) ) ) ).
% add_diff_add
thf(fact_651_add__diff__add,axiom,
! [A3: real,C: real,B3: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_652_sum_Ocl__ivl__Suc,axiom,
! [N3: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_653_sum_Ocl__ivl__Suc,axiom,
! [N3: nat,M2: nat,G: nat > risk_Free_account] :
( ( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= zero_z1425366712893667068ccount ) )
& ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_p1863581527469039996ccount @ ( groups6033208628184776703ccount @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_654_sum_Ocl__ivl__Suc,axiom,
! [N3: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_655_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_656_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_657_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_658_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_659_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_660_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_661_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_662_add__less__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_663_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_664_add__less__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_665_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_666_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_667_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_668_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_669_Suc__less__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_less_eq
thf(fact_670_Suc__mono,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_671_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_672_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_673_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_674_add__less__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_675_add__less__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_676_add__less__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_677_add__less__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_678_less__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_679_less__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_680_less__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_681_less__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_682_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_683_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_684_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_685_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_686_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_687_add__gr__0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_688_zero__less__diff,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M2 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% zero_less_diff
thf(fact_689_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_690_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_691_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_692_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_693_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_694_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_695_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_696_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_697_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_698_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_lift__Suc__mono__less,axiom,
! [F: nat > real,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N3 @ N5 )
=> ( ord_less_real @ ( F @ N3 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_710_lift__Suc__mono__less,axiom,
! [F: nat > nat,N3: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N3 @ N5 )
=> ( ord_less_nat @ ( F @ N3 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_711_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N3: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N3 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_712_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N3: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N3 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N3 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_713_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_714_diff__less__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_715_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_716_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_717_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_718_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_719_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_720_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_721_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_722_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_723_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_724_less__not__refl2,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( M2 != N3 ) ) ).
% less_not_refl2
thf(fact_725_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_726_nat__neq__iff,axiom,
! [M2: nat,N3: nat] :
( ( M2 != N3 )
= ( ( ord_less_nat @ M2 @ N3 )
| ( ord_less_nat @ N3 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_727_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_728_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_729_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_730_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_731_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_732_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_733_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_749_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_759_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_760_order__less__asym_H,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_761_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_762_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_763_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_764_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_765_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_766_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_767_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_768_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_769_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_770_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_771_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_772_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_782_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_783_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_784_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_785_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_786_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_787_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_788_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_789_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_790_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_791_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_792_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_793_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_794_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_795_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_796_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_797_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_798_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_799_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_800_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_801_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_802_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_803_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_804_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_805_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_806_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_807_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_808_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_809_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_810_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_811_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_812_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_813_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_814_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_815_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_816_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_817_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_818_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_819_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_820_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_821_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_822_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_823_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_824_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_843_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_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_865_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_866_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_867_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_868_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_869_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_870_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_871_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_872_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_873_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_874_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_885_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_886_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_887_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_888_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_889_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_928_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_929_gr__implies__not__zero,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_930_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_931_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_932_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_933_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_934_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_935_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_936_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_937_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_938_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_939_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_940_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_941_add__strict__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_942_add__strict__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_943_add__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_944_add__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_945_add__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_946_add__strict__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_947_add__less__imp__less__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_948_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_949_add__less__imp__less__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_950_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_951_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_952_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_953_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_954_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_955_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_956_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_957_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_958_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_959_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_960_gr__implies__not0,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_961_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_962_diff__less,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ) ) ).
% diff_less
thf(fact_963_not__less__less__Suc__eq,axiom,
! [N3: nat,M2: nat] :
( ~ ( ord_less_nat @ N3 @ M2 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M2 ) )
= ( N3 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_964_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_965_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_966_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_967_diff__less__Suc,axiom,
! [M2: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_968_Suc__less__SucD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_969_less__antisym,axiom,
! [N3: nat,M2: nat] :
( ~ ( ord_less_nat @ N3 @ M2 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M2 ) )
=> ( M2 = N3 ) ) ) ).
% less_antisym
thf(fact_970_Suc__less__eq2,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N3 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_971_Suc__diff__Suc,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_972_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N3 ) )
=> ( P @ I ) ) )
= ( ( P @ N3 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N3 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_973_not__less__eq,axiom,
! [M2: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M2 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_974_less__Suc__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_975_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N3 ) )
& ( P @ I ) ) )
= ( ( P @ N3 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N3 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_976_less__SucI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ M2 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_977_less__SucE,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M2 @ N3 )
=> ( M2 = N3 ) ) ) ).
% less_SucE
thf(fact_978_Suc__lessI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ( suc @ M2 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_979_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_980_Suc__lessD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N3 )
=> ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_lessD
thf(fact_981_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_982_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_983_less__imp__le__nat,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_984_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_985_less__or__eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_986_le__neq__implies__less,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( M2 != N3 )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_987_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_988_less__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M2 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_989_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_990_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_991_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_992_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_993_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_994_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_995_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_996_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_997_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N3 ) )
=> ( ord_less_nat @ M2 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_998_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_999_add__diff__inverse__nat,axiom,
! [M2: nat,N3: nat] :
( ~ ( ord_less_nat @ M2 @ N3 )
=> ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M2 @ N3 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1002_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1003_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1004_add__le__less__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1005_add__le__less__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1006_add__less__le__mono,axiom,
! [A3: real,B3: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1007_add__less__le__mono,axiom,
! [A3: nat,B3: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1008_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1009_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1010_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1011_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1012_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_1013_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1014_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1015_add__neg__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1016_add__neg__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1017_add__pos__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1018_add__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1019_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1020_pos__add__strict,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1021_pos__add__strict,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1022_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_1023_less__add__one,axiom,
! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).
% less_add_one
thf(fact_1024_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1025_add__mono1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1026_add__mono1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1027_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: real,B3: real] :
( ~ ( ord_less_real @ A3 @ B3 )
=> ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1028_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ~ ( ord_less_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1029_less__diff__eq,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
= ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).
% less_diff_eq
thf(fact_1030_diff__less__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
= ( ord_less_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1031_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N3 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N3 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1032_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M3: nat] :
( N3
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1033_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N3 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N3 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1034_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M: nat] :
( N3
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_1035_less__Suc__eq__0__disj,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1036_diff__Suc__less,axiom,
! [N3: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I2 ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1037_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1038_Suc__leI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 ) ) ).
% Suc_leI
thf(fact_1039_Suc__le__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_le_eq
thf(fact_1040_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1041_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1042_Suc__le__lessD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_1043_le__less__Suc__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M2 ) )
= ( N3 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1044_less__Suc__eq__le,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_1045_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1046_le__imp__less__Suc,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_nat @ M2 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_1047_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_1048_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 ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1049_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 ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1050_less__natE,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ! [Q2: nat] :
( N3
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1051_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1052_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1053_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1054_less__imp__Suc__add,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ? [K4: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1055_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1056_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_1057_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1058_add__neg__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1059_add__neg__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1060_add__nonneg__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1061_add__nonneg__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1062_add__nonpos__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1063_add__nonpos__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1064_add__pos__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1065_add__pos__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1066_add__strict__increasing,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1067_add__strict__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1068_add__strict__increasing2,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1069_add__strict__increasing2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1070_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1071_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1072_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_1073_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_1074_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_1075_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N3 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1076_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1077_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N3 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1078_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( N3
= ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1079_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1080_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z5: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ? [Y2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y2 ) )
& ! [Z5: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z5 ) )
=> ( ord_less_eq_real @ Y2 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1081_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1082_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1083_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1084_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1085_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_1086_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_1087_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1088_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1089_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 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A2: real,B2: real] :
( ( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A2 ) @ D4 ) )
=> ( P @ A2 @ B2 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1090_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1091_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( 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 @ P2 ) )
= ( groups708209901874060359at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_1092_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups129246275422532515t_real
@ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_real @ ( J = K ) @ one_one_real @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups129246275422532515t_real
@ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_1093_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A )
!= one_one_real )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1102_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups129246275422532515t_real @ G @ A )
!= one_one_real )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1103_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_1104_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_1105_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_1106_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_1107_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_1108_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_nat__descend__induct,axiom,
! [N3: nat,P: nat > $o,M2: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N3 @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1126_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 ) )
& ! [D4: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A3 @ X2 )
& ( ord_less_real @ X2 @ D4 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1127_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 ) )
& ! [D4: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A3 @ X2 )
& ( ord_less_nat @ X2 @ D4 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1128_pinf_I6_J,axiom,
! [T3: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_1129_pinf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_1130_minf_I8_J,axiom,
! [T3: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ~ ( ord_less_eq_real @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_1131_minf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ~ ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_1132_minf_I6_J,axiom,
! [T3: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ord_less_eq_real @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_1133_minf_I6_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_1134_pinf_I8_J,axiom,
! [T3: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_1135_pinf_I8_J,axiom,
! [T3: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_1136_prod_Ocl__ivl__Suc,axiom,
! [N3: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1137_prod_Ocl__ivl__Suc,axiom,
! [N3: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N3 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1138_ln__prod,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( ln_ln_real @ ( F @ X4 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_1139_ln__prod,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( ln_ln_real @ ( F @ X4 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_1140_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1141_add_Oinverse__inverse,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_1142_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_1143_verit__minus__simplify_I4_J,axiom,
! [B3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_1144_mult__is__0,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1145_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1146_mult__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1147_mult__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M2 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1148_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_1149_nat__1__eq__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N3 ) )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1150_nat__mult__eq__1__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1151_finite__atLeastAtMost,axiom,
! [L: nat,U2: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U2 ) ) ).
% finite_atLeastAtMost
thf(fact_1152_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1153_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1154_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1155_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1156_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1157_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1158_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1159_mult__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B3 ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_1160_mult__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_1161_mult__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_1162_mult__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_1163_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_1164_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_1165_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1166_neg__0__equal__iff__equal,axiom,
! [A3: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A3 ) )
= ( zero_z1425366712893667068ccount = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1167_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_1168_neg__equal__0__iff__equal,axiom,
! [A3: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A3 )
= zero_z1425366712893667068ccount )
= ( A3 = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_1169_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_1170_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1171_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1172_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_1173_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_1174_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_1175_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_1176_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_1177_mult__minus__left,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_left
thf(fact_1178_minus__mult__minus,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
= ( times_times_real @ A3 @ B3 ) ) ).
% minus_mult_minus
thf(fact_1179_mult__minus__right,axiom,
! [A3: real,B3: real] :
( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% mult_minus_right
thf(fact_1180_add__minus__cancel,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ ( plus_p1863581527469039996ccount @ ( uminus3377898441596595772ccount @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_1181_add__minus__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ A3 @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 ) )
= B3 ) ).
% add_minus_cancel
thf(fact_1182_minus__add__cancel,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( uminus3377898441596595772ccount @ A3 ) @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_1183_minus__add__cancel,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( plus_plus_real @ A3 @ B3 ) )
= B3 ) ).
% minus_add_cancel
thf(fact_1184_minus__add__distrib,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( plus_p1863581527469039996ccount @ A3 @ B3 ) )
= ( plus_p1863581527469039996ccount @ ( uminus3377898441596595772ccount @ A3 ) @ ( uminus3377898441596595772ccount @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_1185_minus__add__distrib,axiom,
! [A3: real,B3: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) ) ) ).
% minus_add_distrib
thf(fact_1186_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_1187_one__eq__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N3 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1188_mult__eq__1__iff,axiom,
! [M2: nat,N3: nat] :
( ( ( times_times_nat @ M2 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1189_nat__0__less__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1190_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1191_mult__scaleR__left,axiom,
! [A3: real,X: real,Y: real] :
( ( times_times_real @ ( real_V1485227260804924795R_real @ A3 @ X ) @ Y )
= ( real_V1485227260804924795R_real @ A3 @ ( times_times_real @ X @ Y ) ) ) ).
% mult_scaleR_left
thf(fact_1192_mult__scaleR__right,axiom,
! [X: real,A3: real,Y: real] :
( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A3 @ Y ) )
= ( real_V1485227260804924795R_real @ A3 @ ( times_times_real @ X @ Y ) ) ) ).
% mult_scaleR_right
thf(fact_1193_mult__Suc__right,axiom,
! [M2: nat,N3: nat] :
( ( times_times_nat @ M2 @ ( suc @ N3 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N3 ) ) ) ).
% mult_Suc_right
thf(fact_1194_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_1195_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_1196_of__real__mult,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_mult
thf(fact_1197_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_1198_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_1199_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_1200_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_1201_mult__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ( times_times_real @ A3 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1202_mult__cancel__right1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1203_mult__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ( times_times_real @ C @ A3 )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1204_mult__cancel__left1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1205_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_1206_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_1207_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_1208_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_1209_ab__left__minus,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( uminus3377898441596595772ccount @ A3 ) @ A3 )
= zero_z1425366712893667068ccount ) ).
% ab_left_minus
thf(fact_1210_ab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1211_add_Oright__inverse,axiom,
! [A3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A3 @ ( uminus3377898441596595772ccount @ A3 ) )
= zero_z1425366712893667068ccount ) ).
% add.right_inverse
thf(fact_1212_add_Oright__inverse,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1213_diff__0,axiom,
! [A3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ A3 )
= ( uminus3377898441596595772ccount @ A3 ) ) ).
% diff_0
thf(fact_1214_diff__0,axiom,
! [A3: real] :
( ( minus_minus_real @ zero_zero_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ).
% diff_0
thf(fact_1215_verit__minus__simplify_I3_J,axiom,
! [B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ B3 )
= ( uminus3377898441596595772ccount @ B3 ) ) ).
% verit_minus_simplify(3)
thf(fact_1216_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_1217_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_1218_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_1219_uminus__add__conv__diff,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( uminus3377898441596595772ccount @ A3 ) @ B3 )
= ( minus_4846202936726426316ccount @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_1220_uminus__add__conv__diff,axiom,
! [A3: real,B3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B3 )
= ( minus_minus_real @ B3 @ A3 ) ) ).
% uminus_add_conv_diff
thf(fact_1221_diff__minus__eq__add,axiom,
! [A3: risk_Free_account,B3: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A3 @ ( uminus3377898441596595772ccount @ B3 ) )
= ( plus_p1863581527469039996ccount @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_1222_diff__minus__eq__add,axiom,
! [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ ( uminus_uminus_real @ B3 ) )
= ( plus_plus_real @ A3 @ B3 ) ) ).
% diff_minus_eq_add
thf(fact_1223_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_1224_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_1225_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > risk_Free_account] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6033208628184776703ccount @ G @ A )
= zero_z1425366712893667068ccount ) ) ).
% sum.infinite
thf(fact_1226_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_1227_sum_Oinfinite,axiom,
! [A: set_real,G: real > real] :
( ~ ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_1228_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_1229_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_1230_one__le__mult__iff,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N3 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1231_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1232_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_1233_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ N3 ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1234_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N7 )
=> ( ord_less_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1235_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_1236_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_1237_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1238_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1239_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1240_diff__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1241_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_1242_add__mult__distrib,axiom,
! [M2: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N3 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1243_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N3 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% add_mult_distrib2
thf(fact_1244_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1245_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1246_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1247_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1248_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1249_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( M2 = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_1250_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1251_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N7 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1252_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_1253_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_1254_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1255_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1256_mult__Suc,axiom,
! [M2: nat,N3: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N3 )
= ( plus_plus_nat @ N3 @ ( times_times_nat @ M2 @ N3 ) ) ) ).
% mult_Suc
thf(fact_1257_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N3: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1258_mult__eq__self__implies__10,axiom,
! [M2: nat,N3: nat] :
( ( M2
= ( times_times_nat @ M2 @ N3 ) )
=> ( ( N3 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1259_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X4: real,Y5: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1260_one__less__mult,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N3 ) ) ) ) ).
% one_less_mult
thf(fact_1261_n__less__m__mult__n,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ M2 @ N3 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1262_n__less__n__mult__m,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1263_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_1264_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_1265_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_1266_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
% 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,
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ x ) @ ( set_ord_atMost_nat @ n ) )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ y ) @ ( set_ord_atMost_nat @ n ) ) ) ).
%------------------------------------------------------------------------------