TPTP Problem File: SLH0913^1.p
View Solutions
- 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 : Khovanskii_Theorem/0008_Khovanskii/prob_00848_031101__13588296_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1483 ( 402 unt; 207 typ; 0 def)
% Number of atoms : 4550 (1301 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12916 ( 354 ~; 77 |; 349 &;9870 @)
% ( 0 <=>;2266 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 1148 (1148 >; 0 *; 0 +; 0 <<)
% Number of symbols : 191 ( 188 usr; 19 con; 0-3 aty)
% Number of variables : 4202 ( 443 ^;3592 !; 167 ?;4202 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:08.712
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (188)
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
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_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
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_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
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__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4396056296759096172at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
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__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
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__List_Omonoid__add__class_Osum__list_001t__Complex__Ocomplex,type,
groups486868518411355989omplex: list_complex > complex ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
groups4559388385066561235st_int: list_int > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Real__Oreal,type,
groups6723090944982001619t_real: list_real > real ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
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_Khovanskii_OKhovanskii_Oaugmentum,type,
augmentum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Odementum,type,
dementum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olength__sum__set,type,
length_sum_set: nat > nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elements,type,
minimal_elements: set_list_nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elementsp,type,
minimal_elementsp: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_List_Obutlast_001t__Complex__Ocomplex,type,
butlast_complex: list_complex > list_complex ).
thf(sy_c_List_Obutlast_001t__Int__Oint,type,
butlast_int: list_int > list_int ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Real__Oreal,type,
butlast_real: list_real > list_real ).
thf(sy_c_List_Ocan__select_001t__Complex__Ocomplex,type,
can_select_complex: ( complex > $o ) > set_complex > $o ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
distinct_complex: list_complex > $o ).
thf(sy_c_List_Odistinct_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
distinct_real: list_real > $o ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_ONil_001t__Complex__Ocomplex,type,
nil_complex: list_complex ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_Olist__all_001t__Complex__Ocomplex,type,
list_all_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__Nat__Onat_J,type,
list_all_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Real__Oreal,type,
list_all_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
map_complex_complex: ( complex > complex ) > list_complex > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Int__Oint,type,
map_complex_int: ( complex > int ) > list_complex > list_int ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Nat__Onat,type,
map_complex_nat: ( complex > nat ) > list_complex > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_complex_real: ( complex > real ) > list_complex > list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
map_list_nat_int: ( list_nat > int ) > list_list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Int__Oint,type,
map_real_int: ( real > int ) > list_real > list_int ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
map_real_nat: ( real > nat ) > list_real > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist__ex1_001t__Complex__Ocomplex,type,
list_ex1_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist__ex_001t__Complex__Ocomplex,type,
list_ex_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Real__Oreal,type,
list_ex_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
list_update_complex: list_complex > nat > complex > list_complex ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
list_update_real: list_real > nat > real > list_real ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > complex ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__Complex__Ocomplex,type,
removeAll_complex: complex > list_complex > list_complex ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
removeAll_real: real > list_real > list_real ).
thf(sy_c_List_Osorted__wrt_001t__Complex__Ocomplex,type,
sorted_wrt_complex: ( complex > complex > $o ) > list_complex > $o ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $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__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $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__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_nat > $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__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $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_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or4185896845444216793st_nat: list_nat > set_list_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_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Weierstrass__Theorems_OBernstein,type,
weiers7429072931691461095nstein: nat > nat > real > real ).
thf(sy_c_Weierstrass__Theorems_Oreal__polynomial__function_001t__Real__Oreal,type,
weiers3457258110322917882n_real: ( real > real ) > $o ).
thf(sy_c_Wellfounded_OwfP_001t__List__Olist_It__Nat__Onat_J,type,
wfP_list_nat: ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_ns,type,
ns: list_nat ).
% Relevant facts (1265)
thf(fact_0_length__augmentum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( augmentum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_augmentum
thf(fact_1_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_2_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: nat] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_4_atMost__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_5_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_6_sum__delta__notmem_I4_J,axiom,
! [X2: nat,S: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( X2 = Y3 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_7_sum__delta__notmem_I4_J,axiom,
! [X2: nat,S: set_nat,P: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y3: nat] : ( if_real @ ( X2 = Y3 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_8_sum__delta__notmem_I4_J,axiom,
! [X2: complex,S: set_complex,P: complex > complex,Q: complex > complex] :
( ~ ( member_complex @ X2 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [Y3: complex] : ( if_complex @ ( X2 = Y3 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups7754918857620584856omplex @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_9_sum__delta__notmem_I3_J,axiom,
! [X2: nat,S: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( Y3 = X2 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_10_sum__delta__notmem_I3_J,axiom,
! [X2: nat,S: set_nat,P: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y3: nat] : ( if_real @ ( Y3 = X2 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_11_sum__delta__notmem_I3_J,axiom,
! [X2: complex,S: set_complex,P: complex > complex,Q: complex > complex] :
( ~ ( member_complex @ X2 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [Y3: complex] : ( if_complex @ ( Y3 = X2 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S )
= ( groups7754918857620584856omplex @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_12_sum__delta__notmem_I2_J,axiom,
! [X2: nat,S: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( X2 = Y3 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_13_sum__delta__notmem_I2_J,axiom,
! [X2: nat,S: set_nat,P: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y3: nat] : ( if_real @ ( X2 = Y3 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_14_sum__delta__notmem_I2_J,axiom,
! [X2: complex,S: set_complex,P: complex > complex,Q: complex > complex] :
( ~ ( member_complex @ X2 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [Y3: complex] : ( if_complex @ ( X2 = Y3 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups7754918857620584856omplex @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_15_sum__delta__notmem_I1_J,axiom,
! [X2: nat,S: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( Y3 = X2 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_16_sum__delta__notmem_I1_J,axiom,
! [X2: nat,S: set_nat,P: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y3: nat] : ( if_real @ ( Y3 = X2 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_17_sum__delta__notmem_I1_J,axiom,
! [X2: complex,S: set_complex,P: complex > complex,Q: complex > complex] :
( ~ ( member_complex @ X2 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [Y3: complex] : ( if_complex @ ( Y3 = X2 ) @ ( P @ X2 ) @ ( Q @ Y3 ) )
@ S )
= ( groups7754918857620584856omplex @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_18_sum_Oswap,axiom,
! [G: nat > nat > nat,B: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_19_sum_Oswap,axiom,
! [G: nat > nat > real,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_20_sum_Oswap,axiom,
! [G: complex > complex > complex,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I2: complex] : ( groups7754918857620584856omplex @ ( G @ I2 ) @ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I2: complex] : ( G @ I2 @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_21_augmentum_Osimps_I1_J,axiom,
( ( augmentum @ nil_nat )
= nil_nat ) ).
% augmentum.simps(1)
thf(fact_22_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_23_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: nat > real,J2: real > nat,T: set_nat,H: nat > nat,G: real > nat] :
( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_real @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_24_sum_Oreindex__bij__witness,axiom,
! [S2: set_complex,I3: nat > complex,J2: complex > nat,T: set_nat,H: nat > nat,G: complex > nat] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_complex @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_25_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: nat > real,J2: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_real @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_26_sum_Oreindex__bij__witness,axiom,
! [S2: set_complex,I3: nat > complex,J2: complex > nat,T: set_nat,H: nat > real,G: complex > real] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_complex @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_27_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: complex > nat,J2: nat > complex,T: set_complex,H: complex > complex,G: nat > complex] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( member_complex @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( member_nat @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_28_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: complex > real,J2: real > complex,T: set_complex,H: complex > complex,G: real > complex] :
( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( member_complex @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( member_real @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_29_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: real > nat,J2: nat > real,T: set_real,H: real > nat,G: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( member_real @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( member_nat @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_30_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: complex > nat,J2: nat > complex,T: set_complex,H: complex > nat,G: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( member_complex @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( member_nat @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_31_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: nat > nat,J2: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_32_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: real > nat,J2: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( I3 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( member_real @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( ( J2 @ ( I3 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T )
=> ( member_nat @ ( I3 @ B2 ) @ S2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S2 )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_33_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( ( member_real @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_34_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > complex,A: set_complex,H: complex > nat,Gamma: nat > nat,Phi: complex > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( ( member_complex @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_35_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( ( member_real @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_36_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > complex,A: set_complex,H: complex > nat,Gamma: nat > real,Phi: complex > real] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( ( member_complex @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_37_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > nat,A: set_nat,H: nat > complex,Gamma: complex > complex,Phi: nat > complex] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ( ( member_nat @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2073611262835488442omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_38_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > real,A: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ( ( member_real @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_39_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y4: real] :
( ( member_real @ Y4 @ B )
=> ( ( member_nat @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups1935376822645274424al_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_40_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > nat,A: set_nat,H: nat > complex,Gamma: complex > nat,Phi: nat > nat] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ( ( member_nat @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups5693394587270226106ex_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_41_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( ( member_nat @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_42_sum_Oeq__general__inverses,axiom,
! [B: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y4: real] :
( ( member_real @ Y4 @ B )
=> ( ( member_nat @ ( K @ Y4 ) @ A )
& ( ( H @ ( K @ Y4 ) )
= Y4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_43_sum_Oeq__general,axiom,
! [B: set_nat,A: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_44_sum_Oeq__general,axiom,
! [B: set_nat,A: set_complex,H: complex > nat,Gamma: nat > nat,Phi: complex > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_45_sum_Oeq__general,axiom,
! [B: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_46_sum_Oeq__general,axiom,
! [B: set_nat,A: set_complex,H: complex > nat,Gamma: nat > real,Phi: complex > real] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_47_sum_Oeq__general,axiom,
! [B: set_complex,A: set_nat,H: nat > complex,Gamma: complex > complex,Phi: nat > complex] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2073611262835488442omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_48_sum_Oeq__general,axiom,
! [B: set_complex,A: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_49_sum_Oeq__general,axiom,
! [B: set_real,A: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y4: real] :
( ( member_real @ Y4 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups1935376822645274424al_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_50_sum_Oeq__general,axiom,
! [B: set_complex,A: set_nat,H: nat > complex,Gamma: complex > nat,Phi: nat > nat] :
( ! [Y4: complex] :
( ( member_complex @ Y4 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_complex @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups5693394587270226106ex_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_51_sum_Oeq__general,axiom,
! [B: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_52_sum_Oeq__general,axiom,
! [B: set_real,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y4: real] :
( ( member_real @ Y4 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y4 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y4 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_real @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_53_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_54_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_55_sum_Ocong,axiom,
! [A: set_complex,B: set_complex,G: complex > complex,H: complex > complex] :
( ( A = B )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= ( groups7754918857620584856omplex @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_56_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_57_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
& ( P2 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_58_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_59_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_60_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_61_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_62_list__all__length,axiom,
( list_all_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).
% list_all_length
thf(fact_63_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_64_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_65_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_66_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_67_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_68_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ( P @ M ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_69_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_70_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_71_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_72_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_73_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_74_list_Omap__disc__iff,axiom,
! [F: nat > nat,A3: list_nat] :
( ( ( map_nat_nat @ F @ A3 )
= nil_nat )
= ( A3 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_75_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_76_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_77_length__list__update,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_78_list__update__id,axiom,
! [Xs: list_nat,I3: nat] :
( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_79_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_80_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_81_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A3: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A3 @ ( collect_list_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A3: complex,P: complex > $o] :
( ( member_complex @ A3 @ ( collect_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X5: real] : ( member_real @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X5: int] : ( member_int @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X5: complex] : ( member_complex @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_92_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_93_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_94_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_95_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_96_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_97_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_98_sum_Oneutral__const,axiom,
! [A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_99_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_100_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_101_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_102_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_103_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_104_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_105_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_106_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_107_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_108_list__update_Osimps_I1_J,axiom,
! [I3: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I3 @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_109_list__update__code_I1_J,axiom,
! [I3: nat,Y2: nat] :
( ( list_update_nat @ nil_nat @ I3 @ Y2 )
= nil_nat ) ).
% list_update_code(1)
thf(fact_110_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_111_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_112_sum_Oneutral,axiom,
! [A: set_complex,G: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_113_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_114_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > nat,A: set_complex] :
( ( ( groups5693394587270226106ex_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_115_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_116_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_117_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > int,A: set_complex] :
( ( ( groups5690904116761175830ex_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_118_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_119_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A: set_complex] :
( ( ( groups5808333547571424918x_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_120_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A: set_nat] :
( ( ( groups2073611262835488442omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_121_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_122_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_123_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_124_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_125_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_126_nth__list__update,axiom,
! [I3: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_127_list__update__same__conv,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_128_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_129_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_130_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_131_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_132_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_133_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_134_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_135_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_136_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_137_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_138_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_139_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_140_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_141_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_142_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_143_zero__notin__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) ) ) ).
% zero_notin_augmentum
thf(fact_144_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_145_set__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_146_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_147_map__idI,axiom,
! [Xs: list_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_real_real @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_148_map__idI,axiom,
! [Xs: list_complex,F: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_complex_complex @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_149_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_150_list_Omap__ident__strong,axiom,
! [T2: list_list_nat,F: list_nat > list_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_151_list_Omap__ident__strong,axiom,
! [T2: list_real,F: real > real] :
( ! [Z2: real] :
( ( member_real @ Z2 @ ( set_real2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_real_real @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_152_list_Omap__ident__strong,axiom,
! [T2: list_complex,F: complex > complex] :
( ! [Z2: complex] :
( ( member_complex @ Z2 @ ( set_complex2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_complex_complex @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_153_list_Omap__ident__strong,axiom,
! [T2: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_154_list__all__cong,axiom,
! [X2: list_list_nat,Ya2: list_list_nat,P: list_nat > $o,Pa: list_nat > $o] :
( ( X2 = Ya2 )
=> ( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ Ya2 ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_list_nat @ P @ X2 )
= ( list_all_list_nat @ Pa @ Ya2 ) ) ) ) ).
% list_all_cong
thf(fact_155_list__all__cong,axiom,
! [X2: list_real,Ya2: list_real,P: real > $o,Pa: real > $o] :
( ( X2 = Ya2 )
=> ( ! [Z2: real] :
( ( member_real @ Z2 @ ( set_real2 @ Ya2 ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_real @ P @ X2 )
= ( list_all_real @ Pa @ Ya2 ) ) ) ) ).
% list_all_cong
thf(fact_156_list__all__cong,axiom,
! [X2: list_complex,Ya2: list_complex,P: complex > $o,Pa: complex > $o] :
( ( X2 = Ya2 )
=> ( ! [Z2: complex] :
( ( member_complex @ Z2 @ ( set_complex2 @ Ya2 ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_complex @ P @ X2 )
= ( list_all_complex @ Pa @ Ya2 ) ) ) ) ).
% list_all_cong
thf(fact_157_list__all__cong,axiom,
! [X2: list_nat,Ya2: list_nat,P: nat > $o,Pa: nat > $o] :
( ( X2 = Ya2 )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya2 ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_nat @ P @ X2 )
= ( list_all_nat @ Pa @ Ya2 ) ) ) ) ).
% list_all_cong
thf(fact_158_list_Opred__mono__strong,axiom,
! [P: list_nat > $o,X2: list_list_nat,Pa: list_nat > $o] :
( ( list_all_list_nat @ P @ X2 )
=> ( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_list_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_159_list_Opred__mono__strong,axiom,
! [P: real > $o,X2: list_real,Pa: real > $o] :
( ( list_all_real @ P @ X2 )
=> ( ! [Z2: real] :
( ( member_real @ Z2 @ ( set_real2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_real @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_160_list_Opred__mono__strong,axiom,
! [P: complex > $o,X2: list_complex,Pa: complex > $o] :
( ( list_all_complex @ P @ X2 )
=> ( ! [Z2: complex] :
( ( member_complex @ Z2 @ ( set_complex2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_complex @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_161_list_Opred__mono__strong,axiom,
! [P: nat > $o,X2: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P @ X2 )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_162_in__set__butlastD,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_163_in__set__butlastD,axiom,
! [X2: real,Xs: list_real] :
( ( member_real @ X2 @ ( set_real2 @ ( butlast_real @ Xs ) ) )
=> ( member_real @ X2 @ ( set_real2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_164_in__set__butlastD,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ ( butlast_complex @ Xs ) ) )
=> ( member_complex @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_165_in__set__butlastD,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_166_list__ex__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > $o,G: list_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_list_nat @ F @ Xs )
= ( list_ex_list_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_167_list__ex__cong,axiom,
! [Xs: list_real,Ys: list_real,F: real > $o,G: real > $o] :
( ( Xs = Ys )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_real @ F @ Xs )
= ( list_ex_real @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_168_list__ex__cong,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > $o,G: complex > $o] :
( ( Xs = Ys )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_complex @ F @ Xs )
= ( list_ex_complex @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_169_list__ex__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_170_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P2: list_nat > $o,Xs2: list_list_nat] :
? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs2 ) )
& ( P2 @ X5 )
& ! [Y3: list_nat] :
( ( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_171_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P2: real > $o,Xs2: list_real] :
? [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
& ( P2 @ X5 )
& ! [Y3: real] :
( ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_172_list__ex1__iff,axiom,
( list_ex1_complex
= ( ^ [P2: complex > $o,Xs2: list_complex] :
? [X5: complex] :
( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
& ( P2 @ X5 )
& ! [Y3: complex] :
( ( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_173_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X5 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_174_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_175_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_176_length__pos__if__in__set,axiom,
! [X2: real,Xs: list_real] :
( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_177_length__pos__if__in__set,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_178_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_179_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( P @ X5 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_180_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X2: list_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I ) ) )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_181_all__nth__imp__all__set,axiom,
! [Xs: list_real,P: real > $o,X2: real] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ I ) ) )
=> ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_182_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P: complex > $o,X2: complex] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ I ) ) )
=> ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_183_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_184_in__set__conv__nth,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_185_in__set__conv__nth,axiom,
! [X2: real,Xs: list_real] :
( ( member_real @ X2 @ ( set_real2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_186_in__set__conv__nth,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_187_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_188_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_189_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_190_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_191_nth__mem,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_192_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_193_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_194_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_195_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_196_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_197_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_198_set__update__memI,axiom,
! [N: nat,Xs: list_real,X2: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_199_set__update__memI,axiom,
! [N: nat,Xs: list_complex,X2: complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ X2 @ ( set_complex2 @ ( list_update_complex @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_200_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_201_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_202_sum__list__augmentum,axiom,
! [Ns: list_nat] :
( ( member_nat @ ( groups4561878855575611511st_nat @ Ns ) @ ( set_nat2 @ ( augmentum @ Ns ) ) )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ns ) ) ) ).
% sum_list_augmentum
thf(fact_203_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_204_distinct__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( distinct_nat @ ( augmentum @ Ns ) ) ) ).
% distinct_augmentum
thf(fact_205_distinct__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_206_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_207_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_208_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_209_set__conv__nth,axiom,
( set_list_nat2
= ( ^ [Xs2: list_list_nat] :
( collect_list_nat
@ ^ [Uu: list_nat] :
? [I2: nat] :
( ( Uu
= ( nth_list_nat @ Xs2 @ I2 ) )
& ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_210_set__conv__nth,axiom,
( set_int2
= ( ^ [Xs2: list_int] :
( collect_int
@ ^ [Uu: int] :
? [I2: nat] :
( ( Uu
= ( nth_int @ Xs2 @ I2 ) )
& ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_211_set__conv__nth,axiom,
( set_complex2
= ( ^ [Xs2: list_complex] :
( collect_complex
@ ^ [Uu: complex] :
? [I2: nat] :
( ( Uu
= ( nth_complex @ Xs2 @ I2 ) )
& ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_212_set__conv__nth,axiom,
( set_nat2
= ( ^ [Xs2: list_nat] :
( collect_nat
@ ^ [Uu: nat] :
? [I2: nat] :
( ( Uu
= ( nth_nat @ Xs2 @ I2 ) )
& ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_213_length__removeAll__less,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_214_length__removeAll__less,axiom,
! [X2: real,Xs: list_real] :
( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X2 @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_215_length__removeAll__less,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3451745648224563538omplex @ ( removeAll_complex @ X2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_216_length__removeAll__less,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_217_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_218_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_219_removeAll__id,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_220_removeAll__id,axiom,
! [X2: real,Xs: list_real] :
( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( ( removeAll_real @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_221_removeAll__id,axiom,
! [X2: complex,Xs: list_complex] :
( ~ ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( ( removeAll_complex @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_222_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_223_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_224_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri8010041392384452111omplex @ M2 )
= zero_zero_complex )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_225_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_226_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_227_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_228_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_229_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_230_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_231_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_232_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_233_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_234_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_235_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_236_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_237_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_238_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_239_distinct__removeAll,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_240_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_241_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A4: set_nat] :
? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_242_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P2: list_nat > $o,A4: set_list_nat] :
? [X5: list_nat] :
( ( member_list_nat @ X5 @ A4 )
& ( P2 @ X5 )
& ! [Y3: list_nat] :
( ( ( member_list_nat @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_243_can__select__def,axiom,
( can_select_real
= ( ^ [P2: real > $o,A4: set_real] :
? [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 )
& ! [Y3: real] :
( ( ( member_real @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_244_can__select__def,axiom,
( can_select_complex
= ( ^ [P2: complex > $o,A4: set_complex] :
? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 )
& ! [Y3: complex] :
( ( ( member_complex @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_245_distinct__butlast,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( butlast_nat @ Xs ) ) ) ).
% distinct_butlast
thf(fact_246_removeAll_Osimps_I1_J,axiom,
! [X2: nat] :
( ( removeAll_nat @ X2 @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_247_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_248_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_249_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_250_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_251_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_252_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_253_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_254_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_255_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_256_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I2 != J )
=> ( ( nth_nat @ Xs2 @ I2 )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_257_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_258_distinct__Ex1,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_259_distinct__Ex1,axiom,
! [Xs: list_real,X2: real] :
( ( distinct_real @ Xs )
=> ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_260_distinct__Ex1,axiom,
! [Xs: list_complex,X2: complex] :
( ( distinct_complex @ Xs )
=> ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_261_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_262_of__nat__sum,axiom,
! [F: complex > nat,A: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A ) )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( semiri8010041392384452111omplex @ ( F @ X5 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_263_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_264_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( semiri1316708129612266289at_nat @ ( F @ X5 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_265_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( semiri5074537144036343181t_real @ ( F @ X5 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_266_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ns ) )
=> ( X5 = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_267_sum__list_ONil,axiom,
( ( groups4559388385066561235st_int @ nil_int )
= zero_zero_int ) ).
% sum_list.Nil
thf(fact_268_sum__list_ONil,axiom,
( ( groups6723090944982001619t_real @ nil_real )
= zero_zero_real ) ).
% sum_list.Nil
thf(fact_269_sum__list_ONil,axiom,
( ( groups486868518411355989omplex @ nil_complex )
= zero_zero_complex ) ).
% sum_list.Nil
thf(fact_270_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_271_sum_Odistinct__set__conv__list,axiom,
! [Xs: list_nat,G: nat > real] :
( ( distinct_nat @ Xs )
=> ( ( groups6591440286371151544t_real @ G @ ( set_nat2 @ Xs ) )
= ( groups6723090944982001619t_real @ ( map_nat_real @ G @ Xs ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_272_sum_Odistinct__set__conv__list,axiom,
! [Xs: list_complex,G: complex > complex] :
( ( distinct_complex @ Xs )
=> ( ( groups7754918857620584856omplex @ G @ ( set_complex2 @ Xs ) )
= ( groups486868518411355989omplex @ ( map_complex_complex @ G @ Xs ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_273_sum_Odistinct__set__conv__list,axiom,
! [Xs: list_nat,G: nat > nat] :
( ( distinct_nat @ Xs )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_nat2 @ Xs ) )
= ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_274_sum__list__distinct__conv__sum__set,axiom,
! [Xs: list_nat,F: nat > real] :
( ( distinct_nat @ Xs )
=> ( ( groups6723090944982001619t_real @ ( map_nat_real @ F @ Xs ) )
= ( groups6591440286371151544t_real @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_275_sum__list__distinct__conv__sum__set,axiom,
! [Xs: list_complex,F: complex > complex] :
( ( distinct_complex @ Xs )
=> ( ( groups486868518411355989omplex @ ( map_complex_complex @ F @ Xs ) )
= ( groups7754918857620584856omplex @ F @ ( set_complex2 @ Xs ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_276_sum__list__distinct__conv__sum__set,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( distinct_nat @ Xs )
=> ( ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) )
= ( groups3542108847815614940at_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_277_sum__list__strict__mono,axiom,
! [Xs: list_real,F: real > int,G: real > int] :
( ( Xs != nil_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_real_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_real_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_278_sum__list__strict__mono,axiom,
! [Xs: list_complex,F: complex > int,G: complex > int] :
( ( Xs != nil_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_complex_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_complex_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_279_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > int,G: nat > int] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_280_sum__list__strict__mono,axiom,
! [Xs: list_real,F: real > real,G: real > real] :
( ( Xs != nil_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups6723090944982001619t_real @ ( map_real_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_real_real @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_281_sum__list__strict__mono,axiom,
! [Xs: list_complex,F: complex > real,G: complex > real] :
( ( Xs != nil_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups6723090944982001619t_real @ ( map_complex_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_complex_real @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_282_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > real,G: nat > real] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups6723090944982001619t_real @ ( map_nat_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_nat_real @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_283_sum__list__strict__mono,axiom,
! [Xs: list_real,F: real > nat,G: real > nat] :
( ( Xs != nil_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_real_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_real_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_284_sum__list__strict__mono,axiom,
! [Xs: list_complex,F: complex > nat,G: complex > nat] :
( ( Xs != nil_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_complex_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_complex_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_285_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_286_sum__list__strict__mono,axiom,
! [Xs: list_list_nat,F: list_nat > int,G: list_nat > int] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups4559388385066561235st_int @ ( map_list_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_list_nat_int @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_287_distinct__sum__list__conv__Sum,axiom,
! [Xs: list_complex] :
( ( distinct_complex @ Xs )
=> ( ( groups486868518411355989omplex @ Xs )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( set_complex2 @ Xs ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_288_distinct__sum__list__conv__Sum,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( groups4561878855575611511st_nat @ Xs )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_289_augmentum__subset__sum__list,axiom,
! [Ns: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) @ ( set_ord_atMost_nat @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% augmentum_subset_sum_list
thf(fact_290_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_291_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_292_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_293_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_294_atMost__iff,axiom,
! [I3: list_nat,K: list_nat] :
( ( member_list_nat @ I3 @ ( set_or4185896845444216793st_nat @ K ) )
= ( ord_less_eq_list_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_295_atMost__iff,axiom,
! [I3: set_nat,K: set_nat] :
( ( member_set_nat @ I3 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_296_atMost__iff,axiom,
! [I3: int,K: int] :
( ( member_int @ I3 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I3 @ K ) ) ).
% atMost_iff
thf(fact_297_atMost__iff,axiom,
! [I3: real,K: real] :
( ( member_real @ I3 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I3 @ K ) ) ).
% atMost_iff
thf(fact_298_atMost__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_299_atMost__subset__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_300_atMost__subset__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y2 ) )
= ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_301_atMost__subset__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X2 ) @ ( set_ord_atMost_real @ Y2 ) )
= ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_302_atMost__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_303_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_304_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_305_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_306_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_307_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_308_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_309_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_310_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_311_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_312_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_313_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B )
= ( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_314_subset__code_I1_J,axiom,
! [Xs: list_real,B: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
=> ( member_real @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_315_subset__code_I1_J,axiom,
! [Xs: list_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ ( set_complex2 @ Xs ) )
=> ( member_complex @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_316_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_317_member__le__sum__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_318_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I: real] :
( ( member_real @ I @ K2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_319_sum__mono,axiom,
! [K2: set_complex,F: complex > nat,G: complex > nat] :
( ! [I: complex] :
( ( member_complex @ I @ K2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K2 ) @ ( groups5693394587270226106ex_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_320_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I: nat] :
( ( member_nat @ I @ K2 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_321_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I: real] :
( ( member_real @ I @ K2 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_322_sum__mono,axiom,
! [K2: set_complex,F: complex > int,G: complex > int] :
( ! [I: complex] :
( ( member_complex @ I @ K2 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K2 ) @ ( groups5690904116761175830ex_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_323_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I: real] :
( ( member_real @ I @ K2 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_324_sum__mono,axiom,
! [K2: set_complex,F: complex > real,G: complex > real] :
( ! [I: complex] :
( ( member_complex @ I @ K2 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ K2 ) @ ( groups5808333547571424918x_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_325_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I: nat] :
( ( member_nat @ I @ K2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_326_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I: nat] :
( ( member_nat @ I @ K2 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_327_sum__mono,axiom,
! [K2: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [I: list_nat] :
( ( member_list_nat @ I @ K2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ K2 ) @ ( groups4396056296759096172at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_328_atMost__def,axiom,
( set_or4185896845444216793st_nat
= ( ^ [U: list_nat] :
( collect_list_nat
@ ^ [X5: list_nat] : ( ord_less_eq_list_nat @ X5 @ U ) ) ) ) ).
% atMost_def
thf(fact_329_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] : ( ord_less_eq_set_nat @ X5 @ U ) ) ) ) ).
% atMost_def
thf(fact_330_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_eq_int @ X5 @ U ) ) ) ) ).
% atMost_def
thf(fact_331_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_eq_real @ X5 @ U ) ) ) ) ).
% atMost_def
thf(fact_332_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_eq_nat @ X5 @ U ) ) ) ) ).
% atMost_def
thf(fact_333_sum__nonneg,axiom,
! [A: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_334_sum__nonneg,axiom,
! [A: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_335_sum__nonneg,axiom,
! [A: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_336_sum__nonneg,axiom,
! [A: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_337_sum__nonneg,axiom,
! [A: set_complex,F: complex > int] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5690904116761175830ex_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_338_sum__nonneg,axiom,
! [A: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_339_sum__nonneg,axiom,
! [A: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_340_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_341_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_342_sum__nonneg,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4396056296759096172at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_343_sum__nonpos,axiom,
! [A: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_344_sum__nonpos,axiom,
! [A: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_345_sum__nonpos,axiom,
! [A: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_346_sum__nonpos,axiom,
! [A: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_347_sum__nonpos,axiom,
! [A: set_complex,F: complex > int] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_348_sum__nonpos,axiom,
! [A: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_349_sum__nonpos,axiom,
! [A: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_350_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_351_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_352_sum__nonpos,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_353_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_354_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_355_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_356_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ zero_zero_int @ X3 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups4559388385066561235st_int @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_357_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ X3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6723090944982001619t_real @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_358_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_359_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ zero_zero_int @ X3 ) )
=> ( ( ( groups4559388385066561235st_int @ Xs )
= zero_zero_int )
= ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
=> ( X5 = zero_zero_int ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_360_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ zero_zero_real @ X3 ) )
=> ( ( ( groups6723090944982001619t_real @ Xs )
= zero_zero_real )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
=> ( X5 = zero_zero_real ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_361_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ( ( groups4561878855575611511st_nat @ Xs )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( X5 = zero_zero_nat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_362_sum__list__nonpos,axiom,
! [Xs: list_int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_eq_int @ X3 @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ Xs ) @ zero_zero_int ) ) ).
% sum_list_nonpos
thf(fact_363_sum__list__nonpos,axiom,
! [Xs: list_real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ X3 @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ Xs ) @ zero_zero_real ) ) ).
% sum_list_nonpos
thf(fact_364_sum__list__nonpos,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X3 @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ zero_zero_nat ) ) ).
% sum_list_nonpos
thf(fact_365_set__update__subsetI,axiom,
! [Xs: list_list_nat,A: set_list_nat,X2: list_nat,I3: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
=> ( ( member_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I3 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_366_set__update__subsetI,axiom,
! [Xs: list_real,A: set_real,X2: real,I3: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A )
=> ( ( member_real @ X2 @ A )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I3 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_367_set__update__subsetI,axiom,
! [Xs: list_complex,A: set_complex,X2: complex,I3: nat] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
=> ( ( member_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I3 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_368_set__update__subsetI,axiom,
! [Xs: list_nat,A: set_nat,X2: nat,I3: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I3 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_369_sum__list__mono,axiom,
! [Xs: list_real,F: real > int,G: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_real_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_real_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_370_sum__list__mono,axiom,
! [Xs: list_complex,F: complex > int,G: complex > int] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_complex_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_complex_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_371_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > int,G: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_nat_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_372_sum__list__mono,axiom,
! [Xs: list_real,F: real > real,G: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_real_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_real_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_373_sum__list__mono,axiom,
! [Xs: list_complex,F: complex > real,G: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_complex_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_complex_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_374_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > real,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ ( map_nat_real @ F @ Xs ) ) @ ( groups6723090944982001619t_real @ ( map_nat_real @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_375_sum__list__mono,axiom,
! [Xs: list_real,F: real > nat,G: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_real_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_real_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_376_sum__list__mono,axiom,
! [Xs: list_complex,F: complex > nat,G: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_complex_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_complex_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_377_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_378_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > int,G: list_nat > int] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ ( map_list_nat_int @ F @ Xs ) ) @ ( groups4559388385066561235st_int @ ( map_list_nat_int @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_379_sum__list__mono2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) )
=> ( ord_less_eq_int @ ( groups4559388385066561235st_int @ Xs ) @ ( groups4559388385066561235st_int @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_380_sum__list__mono2,axiom,
! [Xs: list_real,Ys: list_real] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I ) @ ( nth_real @ Ys @ I ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ Xs ) @ ( groups6723090944982001619t_real @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_381_sum__list__mono2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_382_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_383_int__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A ) ) ).
% int_sum
thf(fact_384_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_385_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_386_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_less @ Xs @ Ys )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_387_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R: nat,N2: nat] :
( collect_list_nat
@ ^ [X5: list_nat] :
( ( ( size_size_list_nat @ X5 )
= R )
& ( ( groups4561878855575611511st_nat @ X5 )
= N2 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_388_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_389_subsetI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ X3 @ B ) )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% subsetI
thf(fact_390_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ X3 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_391_subsetI,axiom,
! [A: set_complex,B: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( member_complex @ X3 @ B ) )
=> ( ord_le211207098394363844omplex @ A @ B ) ) ).
% subsetI
thf(fact_392_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_393_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_394_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_395_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_396_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_397_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_398_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_399_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_400_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_401_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_402_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_403_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_404_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_405_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_406_list__update__beyond,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
=> ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_407_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_408_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_409_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_410_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_411_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_412_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_413_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_414_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_415_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_416_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_417_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_418_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_419_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_420_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ A4 )
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_421_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ A4 )
@ ^ [X5: real] : ( member_real @ X5 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_422_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A4: set_complex,B4: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X5: complex] : ( member_complex @ X5 @ A4 )
@ ^ [X5: complex] : ( member_complex @ X5 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_423_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_424_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_425_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_426_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_427_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_428_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_429_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_430_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_431_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_432_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_433_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_434_pointwise__less__Nil2,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ X2 @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_435_pointwise__less__Nil,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ nil_nat @ X2 ) ).
% pointwise_less_Nil
thf(fact_436_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_437_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_438_length__removeAll__less__eq,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_439_distinct__n__lists,axiom,
! [Xs: list_nat,N: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_list_nat @ ( n_lists_nat @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_440_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_441_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_442_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_443_order__antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_444_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_445_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_446_linorder__le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_447_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_452_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_453_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_454_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_455_ord__le__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_456_ord__le__eq__subst,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_457_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_458_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_459_ord__eq__le__subst,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_460_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_461_ord__eq__le__subst,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_462_ord__eq__le__subst,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_463_ord__eq__le__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_464_ord__eq__le__subst,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_465_ord__eq__le__subst,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_466_ord__eq__le__subst,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_467_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_468_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_469_linorder__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_470_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_471_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_472_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_473_order__eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_474_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_475_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_476_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_477_order__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_478_order__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_479_order__subst2,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_480_order__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_481_order__subst2,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_482_order__subst2,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_483_order__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_484_order__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_485_order__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_486_order__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_487_order__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_488_order__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_489_order__subst1,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_490_order__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_491_order__subst1,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_492_order__subst1,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_493_order__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_494_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_495_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_496_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_497_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_498_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_499_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_500_antisym,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_501_antisym,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_502_dual__order_Otrans,axiom,
! [B3: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ B3 )
=> ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_503_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_504_dual__order_Otrans,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_505_dual__order_Otrans,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B3 )
=> ( ord_less_eq_real @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_506_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_507_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_508_dual__order_Oantisym,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_509_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_510_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_511_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_512_dual__order_Oeq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_513_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_514_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_515_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_516_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_517_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_518_order__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_519_order__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_520_order__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_521_order_Otrans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_522_order_Otrans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_523_order_Otrans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_524_order_Otrans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_525_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_526_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_527_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_528_order__antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_529_ord__le__eq__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_530_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_531_ord__le__eq__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_532_ord__le__eq__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_533_ord__eq__le__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_534_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_535_ord__eq__le__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( A3 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_536_ord__eq__le__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( A3 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_537_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [X5: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_538_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_539_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_540_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_541_le__cases3,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_542_le__cases3,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_543_le__cases3,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_544_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_545_nle__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_546_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_547_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_548_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_549_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_550_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_551_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_552_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z2: real] :
( ( ord_less_real @ X2 @ Z2 )
& ( ord_less_real @ Z2 @ Y2 ) ) ) ).
% dense
thf(fact_553_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_554_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_555_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_556_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_557_order_Oasym,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order.asym
thf(fact_558_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_559_ord__eq__less__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_560_ord__eq__less__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( A3 = B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_561_ord__eq__less__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( A3 = B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_562_ord__less__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_563_ord__less__eq__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_564_ord__less__eq__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_565_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_566_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_567_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_568_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_569_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_570_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_571_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_572_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_573_dual__order_Oasym,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ~ ( ord_less_int @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_574_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_575_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_576_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_577_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_578_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_579_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_580_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_581_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_582_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_583_order_Ostrict__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_584_order_Ostrict__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_585_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_586_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_587_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_588_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_589_dual__order_Ostrict__trans,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_590_dual__order_Ostrict__trans,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_591_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_592_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_593_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_594_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_595_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_596_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_597_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_598_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_599_linorder__neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_600_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_601_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_602_order__less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_603_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_604_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_605_linorder__neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_606_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_607_order__less__asym_H,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_608_order__less__asym_H,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_609_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_610_order__less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_611_order__less__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_612_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_613_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_614_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_615_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_616_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_617_ord__eq__less__subst,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_618_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_619_ord__eq__less__subst,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_620_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_621_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_622_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_623_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_624_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_625_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_626_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_627_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_628_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_629_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_630_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_631_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_632_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_633_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_634_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_635_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_636_order__less__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_638_order__less__subst1,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_639_order__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_640_order__less__subst1,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_641_order__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_642_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_643_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_644_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_645_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_646_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_647_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_648_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_649_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_650_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_651_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_652_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_653_order__less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_654_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_655_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_656_order__less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_657_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_658_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_659_linorder__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_660_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_661_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_662_order__less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_663_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_664_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_665_order__less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_666_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_667_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_668_order__less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_669_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_670_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_671_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_672_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_673_subset__not__subset__eq,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 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_674_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_675_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_676_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X5: list_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_677_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X5: int] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_678_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X5: complex] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_679_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_680_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_681_subset__trans,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_682_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_683_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_684_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_685_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_686_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_687_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [T3: list_nat] :
( ( member_list_nat @ T3 @ A4 )
=> ( member_list_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_688_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A4 )
=> ( member_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_689_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A4: set_complex,B4: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A4 )
=> ( member_complex @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_690_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_691_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_692_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_693_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ A4 )
=> ( member_list_nat @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_694_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( member_real @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_695_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A4: set_complex,B4: set_complex] :
! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( member_complex @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_696_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( member_nat @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_697_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_698_subsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat @ C2 @ A )
=> ( member_list_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_699_subsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_700_subsetD,axiom,
! [A: set_complex,B: set_complex,C2: complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( member_complex @ C2 @ A )
=> ( member_complex @ C2 @ B ) ) ) ).
% subsetD
thf(fact_701_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_702_in__mono,axiom,
! [A: set_list_nat,B: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat @ X2 @ A )
=> ( member_list_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_703_in__mono,axiom,
! [A: set_real,B: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_704_in__mono,axiom,
! [A: set_complex,B: set_complex,X2: complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( member_complex @ X2 @ A )
=> ( member_complex @ X2 @ B ) ) ) ).
% in_mono
thf(fact_705_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_706_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_707_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_708_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_709_Collect__subset,axiom,
! [A: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_710_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_711_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_712_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_713_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_714_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_715_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_716_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_717_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_718_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_719_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_720_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_721_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_722_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_723_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_724_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_725_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_726_order__less__le__subst2,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_727_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_728_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_729_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_730_order__less__le__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_731_order__less__le__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_732_order__less__le__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_733_order__less__le__subst1,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_734_order__less__le__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_735_order__less__le__subst1,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_736_order__less__le__subst1,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_737_order__less__le__subst1,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_738_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_739_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_740_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_741_order__le__less__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_742_order__le__less__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_743_order__le__less__subst2,axiom,
! [A3: int,B3: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_744_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_745_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_746_order__le__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_747_order__le__less__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_748_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_749_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_750_order__le__less__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_751_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_752_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_753_order__le__less__subst1,axiom,
! [A3: int,F: real > int,B3: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_754_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_755_order__le__less__subst1,axiom,
! [A3: real,F: int > real,B3: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_756_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_757_order__le__less__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_758_order__less__le__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
=> ( ord_less_set_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_759_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_760_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_761_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_762_order__le__less__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z3 )
=> ( ord_less_set_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_763_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_764_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_765_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_766_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_767_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_768_order__neq__le__trans,axiom,
! [A3: int,B3: int] :
( ( A3 != B3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_769_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_770_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_771_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_772_order__le__neq__trans,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_773_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_774_order__less__imp__le,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_775_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_776_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_777_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_778_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_779_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_780_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_781_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_782_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_783_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_784_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_785_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_786_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_787_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_788_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X5: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_789_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y3: nat] :
( ( ord_less_nat @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_790_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y3: int] :
( ( ord_less_int @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_791_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y3: real] :
( ( ord_less_real @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_792_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_793_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_794_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_795_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_796_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_797_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_798_order_Ostrict__implies__order,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_799_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_800_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_801_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_802_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_803_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_804_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ B3 )
=> ( ord_less_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_805_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_806_dual__order_Ostrict__trans2,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_807_dual__order_Ostrict__trans2,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_808_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_set_nat @ C2 @ B3 )
=> ( ord_less_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_809_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_810_dual__order_Ostrict__trans1,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_811_dual__order_Ostrict__trans1,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_812_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_813_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_814_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_815_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_816_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_817_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_818_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_819_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_real @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_820_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z3: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_821_dense__ge__bounded,axiom,
! [Z3: real,X2: real,Y2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_822_order_Ostrict__iff__not,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 ) ) ) ) ).
% order.strict_iff_not
thf(fact_823_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_824_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_825_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_826_order_Ostrict__trans2,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_827_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_828_order_Ostrict__trans2,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_829_order_Ostrict__trans2,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_830_order_Ostrict__trans1,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_831_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_832_order_Ostrict__trans1,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_833_order_Ostrict__trans1,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_834_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_835_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_836_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_837_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_838_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_839_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_840_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_841_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_842_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_843_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_844_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_845_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_846_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y3: nat] :
( ( ord_less_eq_nat @ X5 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_847_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_848_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y3: real] :
( ( ord_less_eq_real @ X5 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_849_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_850_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_851_antisym__conv2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_852_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_853_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_854_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_855_antisym__conv1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_856_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_857_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_858_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_859_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_860_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_861_nless__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_862_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_863_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_864_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_865_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_866_leD,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_867_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_868_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_869_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_870_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U2: list_nat > $o,A5: list_nat] :
? [X5: list_nat] :
( ( A5 = X5 )
& ( U2 @ X5 )
& ! [Y3: list_nat] :
( ( U2 @ Y3 )
=> ~ ( pointwise_less @ Y3 @ X5 ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_871_minimal__elementsp_Ointros,axiom,
! [U3: list_nat > $o,X2: list_nat] :
( ( U3 @ X2 )
=> ( ! [Y4: list_nat] :
( ( U3 @ Y4 )
=> ~ ( pointwise_less @ Y4 @ X2 ) )
=> ( minimal_elementsp @ U3 @ X2 ) ) ) ).
% minimal_elementsp.intros
thf(fact_872_minimal__elementsp_Ocases,axiom,
! [U3: list_nat > $o,A3: list_nat] :
( ( minimal_elementsp @ U3 @ A3 )
=> ~ ( ( U3 @ A3 )
=> ~ ! [Y5: list_nat] :
( ( U3 @ Y5 )
=> ~ ( pointwise_less @ Y5 @ A3 ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_873_minimal__elements_Osimps,axiom,
! [A3: list_nat,U3: set_list_nat] :
( ( member_list_nat @ A3 @ ( minimal_elements @ U3 ) )
= ( ? [X5: list_nat] :
( ( A3 = X5 )
& ( member_list_nat @ X5 @ U3 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U3 )
=> ~ ( pointwise_less @ Y3 @ X5 ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_874_minimal__elements_Ointros,axiom,
! [X2: list_nat,U3: set_list_nat] :
( ( member_list_nat @ X2 @ U3 )
=> ( ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U3 )
=> ~ ( pointwise_less @ Y4 @ X2 ) )
=> ( member_list_nat @ X2 @ ( minimal_elements @ U3 ) ) ) ) ).
% minimal_elements.intros
thf(fact_875_minimal__elements_Ocases,axiom,
! [A3: list_nat,U3: set_list_nat] :
( ( member_list_nat @ A3 @ ( minimal_elements @ U3 ) )
=> ~ ( ( member_list_nat @ A3 @ U3 )
=> ~ ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ U3 )
=> ~ ( pointwise_less @ Y5 @ A3 ) ) ) ) ).
% minimal_elements.cases
thf(fact_876_minimal__elementsp__minimal__elements__eq,axiom,
! [U3: set_list_nat] :
( ( minimal_elementsp
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ U3 ) )
= ( ^ [X5: list_nat] : ( member_list_nat @ X5 @ ( minimal_elements @ U3 ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_877_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U2: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ U2 ) ) ) ) ) ).
% minimal_elements_def
thf(fact_878_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_879_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_880_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_881_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_882_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_883_pred__subset__eq,axiom,
! [R2: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ R2 )
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_884_pred__subset__eq,axiom,
! [R2: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ R2 )
@ ^ [X5: real] : ( member_real @ X5 @ S2 ) )
= ( ord_less_eq_set_real @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_885_pred__subset__eq,axiom,
! [R2: set_complex,S2: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X5: complex] : ( member_complex @ X5 @ R2 )
@ ^ [X5: complex] : ( member_complex @ X5 @ S2 ) )
= ( ord_le211207098394363844omplex @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_886_pred__subset__eq,axiom,
! [R2: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R2 )
@ ^ [X5: nat] : ( member_nat @ X5 @ S2 ) )
= ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_887_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_888_verit__la__disequality,axiom,
! [A3: int,B3: int] :
( ( A3 = B3 )
| ~ ( ord_less_eq_int @ A3 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_889_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_890_verit__comp__simplify1_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_891_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_892_verit__comp__simplify1_I2_J,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_893_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_894_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_895_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_896_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_897_verit__la__generic,axiom,
! [A3: int,X2: int] :
( ( ord_less_eq_int @ A3 @ X2 )
| ( A3 = X2 )
| ( ord_less_eq_int @ X2 @ A3 ) ) ).
% verit_la_generic
thf(fact_898_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_899_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_900_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_901_sorted__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( augmentum @ Ns ) ) ) ).
% sorted_augmentum
thf(fact_902_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_903_distinct__product__lists,axiom,
! [Xss: list_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss ) )
=> ( distinct_nat @ X3 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_904_minimal__elements__set__tuples__finite,axiom,
! [U3: set_list_nat,R3: nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U3 )
=> ( ( size_size_list_nat @ X3 )
= R3 ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U3 ) ) ) ).
% minimal_elements_set_tuples_finite
thf(fact_905_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_906_finite__length__sum__set,axiom,
! [R3: nat,N: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R3 @ N ) ) ).
% finite_length_sum_set
thf(fact_907_List_Ofinite__set,axiom,
! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_908_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_909_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_910_List_Ofinite__set,axiom,
! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% List.finite_set
thf(fact_911_sum_Oinfinite,axiom,
! [A: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_912_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_913_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_914_sum_Oinfinite,axiom,
! [A: set_int,G: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_915_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_916_sum_Oinfinite,axiom,
! [A: set_int,G: int > real] :
( ~ ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_917_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_918_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups2073611262835488442omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_919_sum_Oinfinite,axiom,
! [A: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A )
=> ( ( groups3049146728041665814omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_920_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_921_sum__eq__0__iff,axiom,
! [F2: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ F2 )
=> ( ( ( groups4396056296759096172at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ F2 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_922_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ F2 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_923_sum__eq__0__iff,axiom,
! [F2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ F2 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_924_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ F2 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_925_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_926_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_927_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K4: complex] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K4: complex] : ( if_nat @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_928_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K4: real] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K4: real] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_929_sum_Odelta_H,axiom,
! [S2: set_nat,A3: nat,B3: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_930_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_931_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K4: complex] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K4: complex] : ( if_int @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_932_sum_Odelta_H,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_933_sum_Odelta_H,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K4: int] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K4: int] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_934_sum_Odelta_H,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K4: complex] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K4: complex] : ( if_real @ ( A3 = K4 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_935_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_936_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_937_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K4: complex] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K4: complex] : ( if_nat @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_938_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K4: real] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K4: real] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_939_sum_Odelta,axiom,
! [S2: set_nat,A3: nat,B3: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A3 @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_940_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_941_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K4: complex] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K4: complex] : ( if_int @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_942_sum_Odelta,axiom,
! [S2: set_real,A3: real,B3: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_real @ A3 @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_943_sum_Odelta,axiom,
! [S2: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K4: int] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K4: int] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_944_sum_Odelta,axiom,
! [S2: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K4: complex] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K4: complex] : ( if_real @ ( K4 = A3 ) @ ( B3 @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_945_finite__lists__distinct__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= N )
& ( distinct_list_nat @ Xs2 )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_946_finite__lists__distinct__length__eq,axiom,
! [A: set_int,N: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= N )
& ( distinct_int @ Xs2 )
& ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_947_finite__lists__distinct__length__eq,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= N )
& ( distinct_complex @ Xs2 )
& ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_948_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
& ( distinct_nat @ Xs2 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_949_sorted__wrt__true,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_950_sorted__wrt__mono__rel,axiom,
! [Xs: list_list_nat,P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X3: list_nat,Y4: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat @ Y4 @ ( set_list_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P @ Xs )
=> ( sorted_wrt_list_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_951_sorted__wrt__mono__rel,axiom,
! [Xs: list_real,P: real > real > $o,Q: real > real > $o] :
( ! [X3: real,Y4: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( member_real @ Y4 @ ( set_real2 @ Xs ) )
=> ( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) ) ) )
=> ( ( sorted_wrt_real @ P @ Xs )
=> ( sorted_wrt_real @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_952_sorted__wrt__mono__rel,axiom,
! [Xs: list_complex,P: complex > complex > $o,Q: complex > complex > $o] :
( ! [X3: complex,Y4: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( member_complex @ Y4 @ ( set_complex2 @ Xs ) )
=> ( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) ) ) )
=> ( ( sorted_wrt_complex @ P @ Xs )
=> ( sorted_wrt_complex @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_953_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_954_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_955_finite__list,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [Xs3: list_list_nat] :
( ( set_list_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_956_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_957_finite__list,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [Xs3: list_int] :
( ( set_int2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_958_finite__list,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_959_infinite__Iic,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A3 ) ) ).
% infinite_Iic
thf(fact_960_sorted__wrt__map,axiom,
! [R2: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ R2 @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X5: nat,Y3: nat] : ( R2 @ ( F @ X5 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_961_finite__sorted__distinct__unique,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [X3: list_list_nat] :
( ( ( set_list_nat2 @ X3 )
= A )
& ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ X3 )
& ( distinct_list_nat @ X3 )
& ! [Y5: list_list_nat] :
( ( ( ( set_list_nat2 @ Y5 )
= A )
& ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Y5 )
& ( distinct_list_nat @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_962_finite__sorted__distinct__unique,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [X3: list_nat] :
( ( ( set_nat2 @ X3 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
& ( distinct_nat @ X3 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_963_finite__sorted__distinct__unique,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [X3: list_int] :
( ( ( set_int2 @ X3 )
= A )
& ( sorted_wrt_int @ ord_less_eq_int @ X3 )
& ( distinct_int @ X3 )
& ! [Y5: list_int] :
( ( ( ( set_int2 @ Y5 )
= A )
& ( sorted_wrt_int @ ord_less_eq_int @ Y5 )
& ( distinct_int @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_964_finite__sorted__distinct__unique,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ? [X3: list_real] :
( ( ( set_real2 @ X3 )
= A )
& ( sorted_wrt_real @ ord_less_eq_real @ X3 )
& ( distinct_real @ X3 )
& ! [Y5: list_real] :
( ( ( ( set_real2 @ Y5 )
= A )
& ( sorted_wrt_real @ ord_less_eq_real @ Y5 )
& ( distinct_real @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_965_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_nat,G: real > nat > nat,R2: real > nat > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups1935376822645274424al_nat
@ ^ [X5: real] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups1935376822645274424al_nat
@ ^ [X5: real] : ( G @ X5 @ Y3 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_966_sum_Oswap__restrict,axiom,
! [A: set_int,B: set_nat,G: int > nat > nat,R2: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X5: int] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( G @ X5 @ Y3 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_967_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X5: complex] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups5693394587270226106ex_nat
@ ^ [X5: complex] : ( G @ X5 @ Y3 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_968_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_nat,G: real > nat > real,R2: real > nat > $o] :
( ( finite_finite_real @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8097168146408367636l_real
@ ^ [X5: real] :
( groups6591440286371151544t_real @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y3: nat] :
( groups8097168146408367636l_real
@ ^ [X5: real] : ( G @ X5 @ Y3 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_969_sum_Oswap__restrict,axiom,
! [A: set_int,B: set_nat,G: int > nat > real,R2: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8778361861064173332t_real
@ ^ [X5: int] :
( groups6591440286371151544t_real @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y3: nat] :
( groups8778361861064173332t_real
@ ^ [X5: int] : ( G @ X5 @ Y3 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_970_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_nat,G: complex > nat > real,R2: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5808333547571424918x_real
@ ^ [X5: complex] :
( groups6591440286371151544t_real @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y3: nat] :
( groups5808333547571424918x_real
@ ^ [X5: complex] : ( G @ X5 @ Y3 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_971_sum_Oswap__restrict,axiom,
! [A: set_real,B: set_complex,G: real > complex > complex,R2: real > complex > $o] :
( ( finite_finite_real @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( groups5754745047067104278omplex
@ ^ [X5: real] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups5754745047067104278omplex
@ ^ [X5: real] : ( G @ X5 @ Y3 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_972_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_complex,G: nat > complex > complex,R2: nat > complex > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( groups2073611262835488442omplex
@ ^ [X5: nat] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups2073611262835488442omplex
@ ^ [X5: nat] : ( G @ X5 @ Y3 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_973_sum_Oswap__restrict,axiom,
! [A: set_int,B: set_complex,G: int > complex > complex,R2: int > complex > $o] :
( ( finite_finite_int @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( groups3049146728041665814omplex
@ ^ [X5: int] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y3: complex] :
( ( member_complex @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups7754918857620584856omplex
@ ^ [Y3: complex] :
( groups3049146728041665814omplex
@ ^ [X5: int] : ( G @ X5 @ Y3 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_974_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_real,G: nat > real > nat,R2: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_real @ B )
=> ( ( groups3542108847815614940at_nat
@ ^ [X5: nat] :
( groups1935376822645274424al_nat @ ( G @ X5 )
@ ( collect_real
@ ^ [Y3: real] :
( ( member_real @ Y3 @ B )
& ( R2 @ X5 @ Y3 ) ) ) )
@ A )
= ( groups1935376822645274424al_nat
@ ^ [Y3: real] :
( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( G @ X5 @ Y3 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A )
& ( R2 @ X5 @ Y3 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_975_finite__lists__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs2: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A )
& ( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_976_finite__lists__length__eq,axiom,
! [A: set_int,N: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs2: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A )
& ( ( size_size_list_int @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_977_finite__lists__length__eq,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs2: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A )
& ( ( size_s3451745648224563538omplex @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_978_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ( size_size_list_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_979_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_980_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_981_strict__sorted__imp__sorted,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_982_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_983_sorted0,axiom,
sorted_wrt_int @ ord_less_eq_int @ nil_int ).
% sorted0
thf(fact_984_sorted0,axiom,
sorted_wrt_real @ ord_less_eq_real @ nil_real ).
% sorted0
thf(fact_985_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_986_strict__sorted__simps_I1_J,axiom,
sorted_wrt_int @ ord_less_int @ nil_int ).
% strict_sorted_simps(1)
thf(fact_987_strict__sorted__simps_I1_J,axiom,
sorted_wrt_real @ ord_less_real @ nil_real ).
% strict_sorted_simps(1)
thf(fact_988_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_989_strict__sorted__equal,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_int @ Ys )
=> ( ( ( set_int2 @ Ys )
= ( set_int2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_990_strict__sorted__equal,axiom,
! [Xs: list_real,Ys: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( ( sorted_wrt_real @ ord_less_real @ Ys )
=> ( ( ( set_real2 @ Ys )
= ( set_real2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_991_sum__mono__inv,axiom,
! [F: real > nat,I5: set_real,G: real > nat,I3: real] :
( ( ( groups1935376822645274424al_nat @ F @ I5 )
= ( groups1935376822645274424al_nat @ G @ I5 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_real @ I3 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_992_sum__mono__inv,axiom,
! [F: int > nat,I5: set_int,G: int > nat,I3: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I5 )
= ( groups4541462559716669496nt_nat @ G @ I5 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_993_sum__mono__inv,axiom,
! [F: complex > nat,I5: set_complex,G: complex > nat,I3: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I5 )
= ( groups5693394587270226106ex_nat @ G @ I5 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_994_sum__mono__inv,axiom,
! [F: real > int,I5: set_real,G: real > int,I3: real] :
( ( ( groups1932886352136224148al_int @ F @ I5 )
= ( groups1932886352136224148al_int @ G @ I5 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_real @ I3 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_995_sum__mono__inv,axiom,
! [F: nat > int,I5: set_nat,G: nat > int,I3: nat] :
( ( ( groups3539618377306564664at_int @ F @ I5 )
= ( groups3539618377306564664at_int @ G @ I5 ) )
=> ( ! [I: nat] :
( ( member_nat @ I @ I5 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_nat @ I3 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_996_sum__mono__inv,axiom,
! [F: int > int,I5: set_int,G: int > int,I3: int] :
( ( ( groups4538972089207619220nt_int @ F @ I5 )
= ( groups4538972089207619220nt_int @ G @ I5 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_997_sum__mono__inv,axiom,
! [F: complex > int,I5: set_complex,G: complex > int,I3: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I5 )
= ( groups5690904116761175830ex_int @ G @ I5 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_int @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_998_sum__mono__inv,axiom,
! [F: real > real,I5: set_real,G: real > real,I3: real] :
( ( ( groups8097168146408367636l_real @ F @ I5 )
= ( groups8097168146408367636l_real @ G @ I5 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_real @ I3 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_999_sum__mono__inv,axiom,
! [F: int > real,I5: set_int,G: int > real,I3: int] :
( ( ( groups8778361861064173332t_real @ F @ I5 )
= ( groups8778361861064173332t_real @ G @ I5 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1000_sum__mono__inv,axiom,
! [F: complex > real,I5: set_complex,G: complex > real,I3: complex] :
( ( ( groups5808333547571424918x_real @ F @ I5 )
= ( groups5808333547571424918x_real @ G @ I5 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_real @ ( F @ I ) @ ( G @ I ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1001_sorted__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X5: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1002_sorted__map,axiom,
! [F: nat > int,Xs: list_nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( map_nat_int @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X5: nat,Y3: nat] : ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1003_sorted__map,axiom,
! [F: nat > real,Xs: list_nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( map_nat_real @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X5: nat,Y3: nat] : ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1004_finite__lists__length__le,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs2: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1005_finite__lists__length__le,axiom,
! [A: set_int,N: nat] :
( ( finite_finite_int @ A )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs2: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1006_finite__lists__length__le,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs2: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1007_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1008_finite__maxlen,axiom,
! [M3: set_list_nat] :
( ( finite8100373058378681591st_nat @ M3 )
=> ? [N3: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M3 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1009_finite__distinct__list,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [Xs3: list_list_nat] :
( ( ( set_list_nat2 @ Xs3 )
= A )
& ( distinct_list_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1010_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1011_finite__distinct__list,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ? [Xs3: list_int] :
( ( ( set_int2 @ Xs3 )
= A )
& ( distinct_int @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1012_finite__distinct__list,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ? [Xs3: list_complex] :
( ( ( set_complex2 @ Xs3 )
= A )
& ( distinct_complex @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1013_sum_Ointer__filter,axiom,
! [A: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X5: real] : ( if_nat @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1014_sum_Ointer__filter,axiom,
! [A: set_int,G: int > nat,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( if_nat @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1015_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups5693394587270226106ex_nat
@ ^ [X5: complex] : ( if_nat @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1016_sum_Ointer__filter,axiom,
! [A: set_real,G: real > int,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups1932886352136224148al_int @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups1932886352136224148al_int
@ ^ [X5: real] : ( if_int @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1017_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( if_int @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1018_sum_Ointer__filter,axiom,
! [A: set_int,G: int > int,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( if_int @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1019_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > int,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups5690904116761175830ex_int
@ ^ [X5: complex] : ( if_int @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1020_sum_Ointer__filter,axiom,
! [A: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X5: real] : ( if_real @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1021_sum_Ointer__filter,axiom,
! [A: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( if_real @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1022_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( P @ X5 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X5: complex] : ( if_real @ ( P @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1023_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1024_strict__sorted__iff,axiom,
! [L: list_int] :
( ( sorted_wrt_int @ ord_less_int @ L )
= ( ( sorted_wrt_int @ ord_less_eq_int @ L )
& ( distinct_int @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1025_strict__sorted__iff,axiom,
! [L: list_real] :
( ( sorted_wrt_real @ ord_less_real @ L )
= ( ( sorted_wrt_real @ ord_less_eq_real @ L )
& ( distinct_real @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1026_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1027_sorted__distinct__set__unique,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( distinct_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
=> ( ( distinct_int @ Ys )
=> ( ( ( set_int2 @ Xs )
= ( set_int2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1028_sorted__distinct__set__unique,axiom,
! [Xs: list_real,Ys: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( distinct_real @ Xs )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Ys )
=> ( ( distinct_real @ Ys )
=> ( ( ( set_real2 @ Xs )
= ( set_real2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1029_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P2: nat > nat > $o,Xs2: list_nat] :
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1030_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1031_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1032_sorted__butlast,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ ( butlast_int @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1033_sorted__butlast,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( butlast_real @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1034_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > nat] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A )
= zero_zero_nat )
= ( ! [X5: real] :
( ( member_real @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1035_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1036_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1037_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > int] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ( ( groups1932886352136224148al_int @ F @ A )
= zero_zero_int )
= ( ! [X5: real] :
( ( member_real @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1038_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ A )
= zero_zero_int )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1039_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ A )
= zero_zero_int )
= ( ! [X5: int] :
( ( member_int @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1040_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ A )
= zero_zero_int )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1041_sum__nonneg__eq__0__iff,axiom,
! [A: set_real,F: real > real] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A )
= zero_zero_real )
= ( ! [X5: real] :
( ( member_real @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1042_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > real] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A )
= zero_zero_real )
= ( ! [X5: int] :
( ( member_int @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1043_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A )
= zero_zero_real )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1044_sum__le__included,axiom,
! [S: set_int,T2: set_int,G: int > nat,I3: int > int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S ) @ ( groups4541462559716669496nt_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1045_sum__le__included,axiom,
! [S: set_int,T2: set_complex,G: complex > nat,I3: complex > int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S ) @ ( groups5693394587270226106ex_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1046_sum__le__included,axiom,
! [S: set_complex,T2: set_int,G: int > nat,I3: int > complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S ) @ ( groups4541462559716669496nt_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1047_sum__le__included,axiom,
! [S: set_complex,T2: set_complex,G: complex > nat,I3: complex > complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S ) @ ( groups5693394587270226106ex_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1048_sum__le__included,axiom,
! [S: set_nat,T2: set_nat,G: nat > int,I3: nat > nat,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S ) @ ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1049_sum__le__included,axiom,
! [S: set_nat,T2: set_int,G: int > int,I3: int > nat,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S ) @ ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1050_sum__le__included,axiom,
! [S: set_nat,T2: set_complex,G: complex > int,I3: complex > nat,F: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S ) @ ( groups5690904116761175830ex_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1051_sum__le__included,axiom,
! [S: set_int,T2: set_nat,G: nat > int,I3: nat > int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ S ) @ ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1052_sum__le__included,axiom,
! [S: set_int,T2: set_int,G: int > int,I3: int > int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ S ) @ ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1053_sum__le__included,axiom,
! [S: set_int,T2: set_complex,G: complex > int,I3: complex > int,F: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ S ) @ ( groups5690904116761175830ex_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1054_sum__strict__mono__ex1,axiom,
! [A: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1055_sum__strict__mono__ex1,axiom,
! [A: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A ) @ ( groups5693394587270226106ex_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1056_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1057_sum__strict__mono__ex1,axiom,
! [A: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( groups4538972089207619220nt_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1058_sum__strict__mono__ex1,axiom,
! [A: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A ) @ ( groups5690904116761175830ex_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1059_sum__strict__mono__ex1,axiom,
! [A: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A ) @ ( groups8778361861064173332t_real @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1060_sum__strict__mono__ex1,axiom,
! [A: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A ) @ ( groups5808333547571424918x_real @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1061_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1062_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1063_sum__strict__mono__ex1,axiom,
! [A: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1064_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1065_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1066_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1067_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1068_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1069_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1070_pinf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_1071_pinf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_1072_pinf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_1073_pinf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_1074_pinf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_1075_pinf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_1076_pinf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_1077_pinf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_int @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_1078_pinf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_1079_pinf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_1080_pinf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_int @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_1081_pinf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_1082_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1083_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1084_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1085_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1086_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1087_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1088_minf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_1089_minf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_1090_minf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_1091_minf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_1092_minf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_1093_minf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_1094_minf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_1095_minf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_int @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_1096_minf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_1097_minf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_1098_minf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_int @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_1099_minf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_1100_not__less__Nil,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ nil_nat ) ).
% not_less_Nil
thf(fact_1101_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_1102_sum__nonneg__0,axiom,
! [S: set_real,F: real > nat,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_real @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1103_sum__nonneg__0,axiom,
! [S: set_int,F: int > nat,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_int @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1104_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > nat,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_complex @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1105_sum__nonneg__0,axiom,
! [S: set_real,F: real > int,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups1932886352136224148al_int @ F @ S )
= zero_zero_int )
=> ( ( member_real @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1106_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > int,I3: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I: nat] :
( ( member_nat @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ S )
= zero_zero_int )
=> ( ( member_nat @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1107_sum__nonneg__0,axiom,
! [S: set_int,F: int > int,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S )
= zero_zero_int )
=> ( ( member_int @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1108_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > int,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ S )
= zero_zero_int )
=> ( ( member_complex @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1109_sum__nonneg__0,axiom,
! [S: set_real,F: real > real,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= zero_zero_real )
=> ( ( member_real @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1110_sum__nonneg__0,axiom,
! [S: set_int,F: int > real,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S )
= zero_zero_real )
=> ( ( member_int @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1111_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > real,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S )
= zero_zero_real )
=> ( ( member_complex @ I3 @ S )
=> ( ( F @ I3 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1112_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > nat,B: nat,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= B )
=> ( ( member_real @ I3 @ S )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1113_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > nat,B: nat,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S )
= B )
=> ( ( member_int @ I3 @ S )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1114_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > nat,B: nat,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S )
= B )
=> ( ( member_complex @ I3 @ S )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1115_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > int,B: int,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups1932886352136224148al_int @ F @ S )
= B )
=> ( ( member_real @ I3 @ S )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1116_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > int,B: int,I3: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I: nat] :
( ( member_nat @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ S )
= B )
=> ( ( member_nat @ I3 @ S )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1117_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > int,B: int,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S )
= B )
=> ( ( member_int @ I3 @ S )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1118_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > int,B: int,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ S )
= B )
=> ( ( member_complex @ I3 @ S )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1119_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > real,B: real,I3: real] :
( ( finite_finite_real @ S )
=> ( ! [I: real] :
( ( member_real @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= B )
=> ( ( member_real @ I3 @ S )
=> ( ord_less_eq_real @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1120_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > real,B: real,I3: int] :
( ( finite_finite_int @ S )
=> ( ! [I: int] :
( ( member_int @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S )
= B )
=> ( ( member_int @ I3 @ S )
=> ( ord_less_eq_real @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1121_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > real,B: real,I3: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I: complex] :
( ( member_complex @ I @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S )
= B )
=> ( ( member_complex @ I3 @ S )
=> ( ord_less_eq_real @ ( F @ I3 ) @ B ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1122_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1123_sorted__iff__nth__mono__less,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1124_sorted__iff__nth__mono__less,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1125_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I3: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I3 @ ( nth_nat @ Ns @ I3 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1126_sum__pos2,axiom,
! [I5: set_real,I3: real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I3 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1127_sum__pos2,axiom,
! [I5: set_int,I3: int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I3 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1128_sum__pos2,axiom,
! [I5: set_complex,I3: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1129_sum__pos2,axiom,
! [I5: set_real,I3: real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I3 @ I5 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I3 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1130_sum__pos2,axiom,
! [I5: set_nat,I3: nat,F: nat > int] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I3 @ I5 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I3 ) )
=> ( ! [I: nat] :
( ( member_nat @ I @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1131_sum__pos2,axiom,
! [I5: set_int,I3: int,F: int > int] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I3 @ I5 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I3 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1132_sum__pos2,axiom,
! [I5: set_complex,I3: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I3 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups5690904116761175830ex_int @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1133_sum__pos2,axiom,
! [I5: set_real,I3: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I3 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
=> ( ! [I: real] :
( ( member_real @ I @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1134_sum__pos2,axiom,
! [I5: set_int,I3: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I3 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
=> ( ! [I: int] :
( ( member_int @ I @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1135_sum__pos2,axiom,
! [I5: set_complex,I3: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
=> ( ! [I: complex] :
( ( member_complex @ I @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1136_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1137_sorted__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1138_sorted__iff__nth__mono,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I2 ) @ ( nth_real @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1139_sorted__nth__mono,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1140_sorted__nth__mono,axiom,
! [Xs: list_int,I3: nat,J2: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1141_sorted__nth__mono,axiom,
! [Xs: list_real,I3: nat,J2: nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1142_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1143_minf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_1144_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1145_dementum__nonzero,axiom,
! [Ns: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( dementum @ Ns ) ) ) ) ) ).
% dementum_nonzero
thf(fact_1146_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1147_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1148_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1149_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A3 @ I2 )
& ( ord_less_eq_int @ I2 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_1150_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A3 @ I2 )
& ( ord_less_int @ I2 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_1151_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A3 @ I2 )
& ( ord_less_eq_int @ I2 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_1152_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A3 @ I2 )
& ( ord_less_int @ I2 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_1153_dementum__Nil,axiom,
( ( dementum @ nil_nat )
= nil_nat ) ).
% dementum_Nil
thf(fact_1154_length__dementum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( dementum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_dementum
thf(fact_1155_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1156_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1157_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_nat @ X5 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1158_finite__less__ub,axiom,
! [F: nat > nat,U4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U4 ) ) ) ) ).
% finite_less_ub
thf(fact_1159_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less_nat @ K4 @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1160_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S3: set_nat] :
? [K4: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_atMost_nat @ K4 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_1161_WFP,axiom,
wfP_list_nat @ pointwise_less ).
% WFP
thf(fact_1162_finite__atLeastLessThan,axiom,
! [L: nat,U4: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U4 ) ) ).
% finite_atLeastLessThan
thf(fact_1163_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X5 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1164_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X5 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1165_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1166_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1167_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1168_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
& ( member_nat @ N6 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1169_finite__atLeastLessThan__int,axiom,
! [L: int,U4: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U4 ) ) ).
% finite_atLeastLessThan_int
thf(fact_1170_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1171_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1172_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1173_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1174_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1175_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1176_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1177_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1178_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1179_mult__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1180_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1181_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1182_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1183_finite__atLeastZeroLessThan__int,axiom,
! [U4: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U4 ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_1184_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1185_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1186_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1187_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1188_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B3: nat > nat] :
( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( A3 @ I ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( B3 @ J3 ) @ ( B3 @ I ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( A3 @ I2 ) @ ( B3 @ I2 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_1189_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1190_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1191_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C2 ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1192_zmult__zless__mono2,axiom,
! [I3: int,J2: int,K: int] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I3 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1193_zmult__zless__mono2__lemma,axiom,
! [I3: int,J2: int,K: nat] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1194_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1195_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1196_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1197_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1198_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_1199_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y3: real] :
( ( ord_less_real @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1200_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y4 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1201_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M2 ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M2 @ N6 )
=> ( ord_less_real @ ( D @ N6 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1202_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1203_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1204_nat__power__less__imp__less,axiom,
! [I3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M2 ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1205_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_1206_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1207_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1208_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1209_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_1210_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1211_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1212_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2 != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_1213_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_1214_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_1215_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1216_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1217_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1218_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1219_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1220_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1221_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1222_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1223_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1224_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1225_less__mult__imp__div__less,axiom,
! [M2: nat,I3: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I3 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I3 ) ) ).
% less_mult_imp_div_less
thf(fact_1226_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1227_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1228_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1229_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1230_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1231_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1232_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I3: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I3 @ K ) )
= ( ord_less_eq_int @ K @ I3 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1233_div__nonpos__pos__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1234_div__nonneg__neg__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1235_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1236_zdiv__mono2__neg,axiom,
! [A3: int,B6: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B6 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1237_zdiv__mono1__neg,axiom,
! [A3: int,A6: int,B3: int] :
( ( ord_less_eq_int @ A3 @ A6 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B3 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1238_zdiv__eq__0__iff,axiom,
! [I3: int,K: int] :
( ( ( divide_divide_int @ I3 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I3 )
& ( ord_less_int @ I3 @ K ) )
| ( ( ord_less_eq_int @ I3 @ zero_zero_int )
& ( ord_less_int @ K @ I3 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1239_zdiv__mono2,axiom,
! [A3: int,B6: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A3 @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1240_zdiv__mono1,axiom,
! [A3: int,A6: int,B3: int] :
( ( ord_less_eq_int @ A3 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A6 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_1241_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1242_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1243_real__polynomial__function__length__sum__set,axiom,
! [R3: nat] :
? [P5: real > real] :
( ( weiers3457258110322917882n_real @ P5 )
& ! [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
=> ( ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ ( length_sum_set @ R3 @ N6 ) ) )
= ( P5 @ ( semiri5074537144036343181t_real @ N6 ) ) ) ) ) ).
% real_polynomial_function_length_sum_set
thf(fact_1244_real__polynomial__function__sum__of__powers,axiom,
! [J2: nat] :
? [P5: real > real] :
( ( weiers3457258110322917882n_real @ P5 )
& ! [N6: nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( power_power_real @ ( semiri5074537144036343181t_real @ I2 ) @ J2 )
@ ( set_ord_atMost_nat @ N6 ) )
= ( P5 @ ( semiri5074537144036343181t_real @ N6 ) ) ) ) ).
% real_polynomial_function_sum_of_powers
thf(fact_1245_real__polynomial__function__imp__sum,axiom,
! [F: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ? [A2: nat > real,N3: nat] :
( F
= ( ^ [X5: real] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( times_times_real @ ( A2 @ I2 ) @ ( power_power_real @ X5 @ I2 ) )
@ ( set_ord_atMost_nat @ N3 ) ) ) ) ) ).
% real_polynomial_function_imp_sum
thf(fact_1246_real__polynomial__function__iff__sum,axiom,
( weiers3457258110322917882n_real
= ( ^ [F3: real > real] :
? [A5: nat > real,N2: nat] :
( F3
= ( ^ [X5: real] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( times_times_real @ ( A5 @ I2 ) @ ( power_power_real @ X5 @ I2 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ) ).
% real_polynomial_function_iff_sum
thf(fact_1247_sum__k__Bernstein,axiom,
! [N: nat,X2: real] :
( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ K4 ) @ ( weiers7429072931691461095nstein @ N @ K4 @ X2 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) ).
% sum_k_Bernstein
thf(fact_1248_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1249_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1250_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1251_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1252_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1253_sum__Bernstein,axiom,
! [N: nat,X2: real] :
( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( weiers7429072931691461095nstein @ N @ K4 @ X2 )
@ ( set_ord_atMost_nat @ N ) )
= one_one_real ) ).
% sum_Bernstein
thf(fact_1254_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1255_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1256_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1257_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1258_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1259_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1260_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1261_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1262_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1263_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1264_Bernstein__nonneg,axiom,
! [X2: real,N: nat,K: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( weiers7429072931691461095nstein @ N @ K @ X2 ) ) ) ) ).
% Bernstein_nonneg
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y2: real] :
( ( if_real @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y2: real] :
( ( if_real @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y2: complex] :
( ( if_complex @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y2: complex] :
( ( if_complex @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
ord_less_nat @ i @ ( size_size_list_nat @ ns ) ).
thf(conj_1,conjecture,
( ( nth_nat @ ( augmentum @ ns ) @ i )
= ( groups3542108847815614940at_nat @ ( nth_nat @ ns ) @ ( set_ord_atMost_nat @ i ) ) ) ).
%------------------------------------------------------------------------------