TPTP Problem File: SLH0111^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0034_Incidence_Matrices/prob_01843_078912__28070214_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1452 ( 681 unt; 178 typ; 0 def)
% Number of atoms : 3600 (2010 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12280 ( 410 ~; 69 |; 343 &;10175 @)
% ( 0 <=>;1283 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 1145 (1145 >; 0 *; 0 +; 0 <<)
% Number of symbols : 164 ( 161 usr; 14 con; 0-4 aty)
% Number of variables : 3888 ( 656 ^;3159 !; 73 ?;3888 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:48:48.920
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
vec_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Real__Oreal_J,type,
vec_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (161)
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
semiri2265585572941072030t_real: nat > real ).
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__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_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_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Int__Oint_J,type,
plus_plus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Int__Oint_J,type,
plus_plus_vec_int: vec_int > vec_int > vec_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
plus_plus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Real__Oreal_J,type,
plus_plus_vec_real: vec_real > vec_real > vec_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
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__Matrix__Omat_It__Int__Oint_J,type,
times_times_mat_int: mat_int > mat_int > mat_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__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_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_Incidence__Matrices_Onon__empty__col_001t__Int__Oint,type,
incide6851923868969248411ol_int: mat_int > nat > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Int__Oint,type,
incide294466202882093137at_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Complex__Ocomplex,type,
incide5998224313882735548omplex: mat_complex > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Int__Oint,type,
incide4964164200581851450ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Nat__Onat,type,
incide4966654671090901726ix_nat: mat_nat > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Real__Oreal,type,
incide4475037519619858106x_real: mat_real > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Complex__Ocomplex,type,
incide7996601054137363008omplex: vec_complex > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Int__Oint,type,
incide3973235006681262014ck_int: vec_int > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Nat__Onat,type,
incide3975725477190312290ck_nat: vec_nat > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Real__Oreal,type,
incide970706021007448894k_real: vec_real > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__int,type,
incide8301514189696901506ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__ring__1_001t__Int__Oint,type,
incide6080938071136783841_1_int: mat_int > $o ).
thf(sy_c_Macaulay__Matrix_Omult__vec__mat_001t__Int__Oint,type,
macaul1993098418559423895at_int: vec_int > mat_int > vec_int ).
thf(sy_c_Matrix_Ocol_001t__Int__Oint,type,
col_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001t__Complex__Ocomplex,type,
dim_row_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Complex__Ocomplex,type,
dim_vec_complex: vec_complex > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Int__Oint,type,
dim_vec_int: vec_int > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Nat__Onat,type,
dim_vec_nat: vec_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Real__Oreal,type,
dim_vec_real: vec_real > nat ).
thf(sy_c_Matrix_Omat__of__row_001t__Int__Oint,type,
mat_of_row_int: vec_int > mat_int ).
thf(sy_c_Matrix_Omat__of__row__fun_001t__Int__Oint,type,
mat_of_row_fun_int: nat > nat > ( nat > vec_int ) > mat_int ).
thf(sy_c_Matrix_Orow_001t__Complex__Ocomplex,type,
row_complex: mat_complex > nat > vec_complex ).
thf(sy_c_Matrix_Orow_001t__Int__Oint,type,
row_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Orow_001t__Nat__Onat,type,
row_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Orow_001t__Real__Oreal,type,
row_real: mat_real > nat > vec_real ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix_Ounit__vec_001t__Complex__Ocomplex,type,
unit_vec_complex: nat > nat > vec_complex ).
thf(sy_c_Matrix_Ounit__vec_001t__Int__Oint,type,
unit_vec_int: nat > nat > vec_int ).
thf(sy_c_Matrix_Ounit__vec_001t__Nat__Onat,type,
unit_vec_nat: nat > nat > vec_nat ).
thf(sy_c_Matrix_Ounit__vec_001t__Real__Oreal,type,
unit_vec_real: nat > nat > vec_real ).
thf(sy_c_Matrix_Oupdate__mat_001t__Int__Oint,type,
update_mat_int: mat_int > product_prod_nat_nat > int > mat_int ).
thf(sy_c_Matrix_Oupdate__vec_001t__Int__Oint,type,
update_vec_int: vec_int > nat > int > vec_int ).
thf(sy_c_Matrix_Ovec__index_001t__Complex__Ocomplex,type,
vec_index_complex: vec_complex > nat > complex ).
thf(sy_c_Matrix_Ovec__index_001t__Int__Oint,type,
vec_index_int: vec_int > nat > int ).
thf(sy_c_Matrix_Ovec__index_001t__Nat__Onat,type,
vec_index_nat: vec_nat > nat > nat ).
thf(sy_c_Matrix_Ovec__index_001t__Real__Oreal,type,
vec_index_real: vec_real > nat > real ).
thf(sy_c_Matrix_Ovec__set_001t__Int__Oint,type,
vec_set_int: vec_int > set_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Int__Oint,type,
matrix8485685120660989714at_int: nat > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Complex__Ocomplex,type,
matrix5128154353246082568omplex: nat > vec_complex ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Int__Oint,type,
matrix2748772424961467270ec_int: nat > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Nat__Onat,type,
matrix2751262895470517546ec_nat: nat > vec_nat ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Real__Oreal,type,
matrix5166576126360777478c_real: nat > vec_real ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Complex__Ocomplex,type,
matrix7538134685299346620omplex: vec_complex > complex ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Int__Oint,type,
matrix3634415343793898042ec_int: vec_int > int ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Nat__Onat,type,
matrix3636905814302948318ec_nat: vec_nat > nat ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Real__Oreal,type,
matrix1363837090280519610c_real: vec_real > real ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Int__Oint,type,
matrix323868623736973467nt_int: mat_int > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001t__Int__Oint_001t__Int__Oint,type,
matrix8301520909418075407nt_int: vec_int > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Complex__Ocomplex_001t__Int__Oint,type,
matrix4488594806643890280ex_int: complex > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Complex__Ocomplex_001t__Nat__Onat,type,
matrix4491085277152940556ex_nat: complex > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Complex__Ocomplex,type,
matrix1846837417924380264omplex: int > complex ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Int__Oint,type,
matrix1697308990001484774nt_int: int > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Nat__Onat,type,
matrix1699799460510535050nt_nat: int > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Real__Oreal,type,
matrix1706393078865277798t_real: int > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Complex__Ocomplex,type,
matrix871301952718202892omplex: nat > complex ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Int__Oint,type,
matrix697955278100430218at_int: nat > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Nat__Onat,type,
matrix700445748609480494at_nat: nat > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Real__Oreal,type,
matrix8742843541027031818t_real: nat > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Int__Oint,type,
matrix4084289606792104422al_int: real > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Nat__Onat,type,
matrix4086780077301154698al_nat: real > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > 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__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > 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__Int__Oint,type,
power_power_int: int > nat > int ).
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__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__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_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_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
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_OlessThan_001t__Complex__Ocomplex,type,
set_or7194820819169546315omplex: complex > set_complex ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__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__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_v_M,type,
m: mat_int ).
thf(sy_v_i,type,
i: nat ).
% Relevant facts (1264)
thf(fact_0_zero__one__matrix__int__axioms,axiom,
incide8301514189696901506ix_int @ m ).
% zero_one_matrix_int_axioms
thf(fact_1_zero__one__matrix__ring__1__axioms,axiom,
incide6080938071136783841_1_int @ m ).
% zero_one_matrix_ring_1_axioms
thf(fact_2_sum_Oneutral__const,axiom,
! [A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [Uu: int] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_3_sum_Oneutral__const,axiom,
! [A: set_int] :
( ( groups3049146728041665814omplex
@ ^ [Uu: int] : zero_zero_complex
@ A )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_4_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [Uu: nat] : zero_zero_complex
@ A )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_5_sum_Oneutral__const,axiom,
! [A: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu: int] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_6_sum_Oneutral__const,axiom,
! [A: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [Uu: int] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_7_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [Uu: nat] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_8_sum_Oneutral__const,axiom,
! [A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_9_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_10_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_11_sum_Oneutral,axiom,
! [A: set_int,G: int > real] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( G @ X )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_12_sum_Oneutral,axiom,
! [A: set_int,G: int > complex] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( G @ X )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_13_sum_Oneutral,axiom,
! [A: set_nat,G: nat > complex] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( G @ X )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_14_sum_Oneutral,axiom,
! [A: set_int,G: int > int] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_15_sum_Oneutral,axiom,
! [A: set_int,G: int > nat] :
( ! [X: int] :
( ( member_int @ X @ A )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_16_sum_Oneutral,axiom,
! [A: set_nat,G: nat > int] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_17_sum_Oneutral,axiom,
! [A: set_complex,G: complex > complex] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( G @ X )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_18_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( G @ X )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_19_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_20_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_21_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > int,A: set_int] :
( ( ( groups4538972089207619220nt_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_22_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_23_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A: set_int] :
( ( ( groups3049146728041665814omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_24_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: int] :
( ( member_int @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_25_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_26_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A )
!= zero_zero_complex )
=> ~ ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_27_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_28_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_29_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_30_zero__one__matrix__axioms,axiom,
incide4964164200581851450ix_int @ m ).
% zero_one_matrix_axioms
thf(fact_31_sum_Oswap,axiom,
! [G: nat > nat > int,B: set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups3539618377306564664at_int @ ( G @ I ) @ B )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_32_sum_Oswap,axiom,
! [G: complex > complex > complex,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_33_sum_Oswap,axiom,
! [G: nat > nat > real,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_34_sum_Oswap,axiom,
! [G: nat > nat > nat,B: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_35_sum_Oswap,axiom,
! [G: nat > int > int,B: set_int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups4538972089207619220nt_int @ ( G @ I ) @ B )
@ A )
= ( groups4538972089207619220nt_int
@ ^ [J: int] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_36_sum_Oswap,axiom,
! [G: complex > int > complex,B: set_int,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups3049146728041665814omplex @ ( G @ I ) @ B )
@ A )
= ( groups3049146728041665814omplex
@ ^ [J: int] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_37_sum_Oswap,axiom,
! [G: complex > nat > complex,B: set_nat,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups2073611262835488442omplex @ ( G @ I ) @ B )
@ A )
= ( groups2073611262835488442omplex
@ ^ [J: nat] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_38_sum_Oswap,axiom,
! [G: nat > int > real,B: set_int,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups8778361861064173332t_real @ ( G @ I ) @ B )
@ A )
= ( groups8778361861064173332t_real
@ ^ [J: int] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_39_sum_Oswap,axiom,
! [G: nat > int > nat,B: set_int,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups4541462559716669496nt_nat @ ( G @ I ) @ B )
@ A )
= ( groups4541462559716669496nt_nat
@ ^ [J: int] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_40_sum_Oswap,axiom,
! [G: int > nat > real,B: set_nat,A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [I: int] : ( groups6591440286371151544t_real @ ( G @ I ) @ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups8778361861064173332t_real
@ ^ [I: int] : ( G @ I @ J )
@ A )
@ B ) ) ).
% sum.swap
thf(fact_41_sum__reorder__triple,axiom,
! [G: nat > nat > nat > int,C: set_nat,B: set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [L: nat] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups3539618377306564664at_int @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups3539618377306564664at_int
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_42_sum__reorder__triple,axiom,
! [G: complex > complex > complex > complex,C: set_complex,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [L: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [L: complex] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_43_sum__reorder__triple,axiom,
! [G: nat > nat > nat > real,C: set_nat,B: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [L: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_44_sum__reorder__triple,axiom,
! [G: nat > nat > nat > nat,C: set_nat,B: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [L: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_45_sum__reorder__triple,axiom,
! [G: nat > nat > int > int,C: set_int,B: set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [L: nat] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups4538972089207619220nt_int @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups4538972089207619220nt_int
@ ^ [J: int] :
( groups3539618377306564664at_int
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_46_sum__reorder__triple,axiom,
! [G: nat > int > nat > int,C: set_nat,B: set_int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [L: nat] :
( groups4538972089207619220nt_int
@ ^ [I: int] : ( groups3539618377306564664at_int @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups4538972089207619220nt_int
@ ^ [I: int] :
( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups3539618377306564664at_int
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_47_sum__reorder__triple,axiom,
! [G: nat > int > int > int,C: set_int,B: set_int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [L: nat] :
( groups4538972089207619220nt_int
@ ^ [I: int] : ( groups4538972089207619220nt_int @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups4538972089207619220nt_int
@ ^ [I: int] :
( groups4538972089207619220nt_int
@ ^ [J: int] :
( groups3539618377306564664at_int
@ ^ [L: nat] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_48_sum__reorder__triple,axiom,
! [G: complex > complex > int > complex,C: set_int,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [L: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups3049146728041665814omplex @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups3049146728041665814omplex
@ ^ [J: int] :
( groups7754918857620584856omplex
@ ^ [L: complex] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_49_sum__reorder__triple,axiom,
! [G: complex > complex > nat > complex,C: set_nat,B: set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [L: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups2073611262835488442omplex @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups2073611262835488442omplex
@ ^ [J: nat] :
( groups7754918857620584856omplex
@ ^ [L: complex] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_50_sum__reorder__triple,axiom,
! [G: complex > int > complex > complex,C: set_complex,B: set_int,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [L: complex] :
( groups3049146728041665814omplex
@ ^ [I: int] : ( groups7754918857620584856omplex @ ( G @ L @ I ) @ C )
@ B )
@ A )
= ( groups3049146728041665814omplex
@ ^ [I: int] :
( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [L: complex] : ( G @ L @ I @ J )
@ A )
@ C )
@ B ) ) ).
% sum_reorder_triple
thf(fact_51_sum__vec__def,axiom,
( matrix1363837090280519610c_real
= ( ^ [V: vec_real] : ( groups6591440286371151544t_real @ ( vec_index_real @ V ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_real @ V ) ) ) ) ) ).
% sum_vec_def
thf(fact_52_sum__vec__def,axiom,
( matrix3636905814302948318ec_nat
= ( ^ [V: vec_nat] : ( groups3542108847815614940at_nat @ ( vec_index_nat @ V ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_nat @ V ) ) ) ) ) ).
% sum_vec_def
thf(fact_53_sum__vec__def,axiom,
( matrix7538134685299346620omplex
= ( ^ [V: vec_complex] : ( groups2073611262835488442omplex @ ( vec_index_complex @ V ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_complex @ V ) ) ) ) ) ).
% sum_vec_def
thf(fact_54_sum__vec__def,axiom,
( matrix3634415343793898042ec_int
= ( ^ [V: vec_int] : ( groups3539618377306564664at_int @ ( vec_index_int @ V ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_int @ V ) ) ) ) ) ).
% sum_vec_def
thf(fact_55_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > int,H: nat > int] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_56_sum_Ocong,axiom,
! [A: set_complex,B: set_complex,G: complex > complex,H: complex > complex] :
( ( A = B )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A )
= ( groups7754918857620584856omplex @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_57_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > real,H: nat > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_58_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_59_sum_Ocong,axiom,
! [A: set_int,B: set_int,G: int > real,H: int > real] :
( ( A = B )
=> ( ! [X: int] :
( ( member_int @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups8778361861064173332t_real @ G @ A )
= ( groups8778361861064173332t_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_60_sum_Ocong,axiom,
! [A: set_int,B: set_int,G: int > complex,H: int > complex] :
( ( A = B )
=> ( ! [X: int] :
( ( member_int @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3049146728041665814omplex @ G @ A )
= ( groups3049146728041665814omplex @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_61_sum_Ocong,axiom,
! [A: set_nat,B: set_nat,G: nat > complex,H: nat > complex] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups2073611262835488442omplex @ G @ A )
= ( groups2073611262835488442omplex @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_62_sum_Ocong,axiom,
! [A: set_int,B: set_int,G: int > int,H: int > int] :
( ( A = B )
=> ( ! [X: int] :
( ( member_int @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= ( groups4538972089207619220nt_int @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_63_sum_Ocong,axiom,
! [A: set_int,B: set_int,G: int > nat,H: int > nat] :
( ( A = B )
=> ( ! [X: int] :
( ( member_int @ X @ B )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= ( groups4541462559716669496nt_nat @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_64_zero__one__matrix__int_Ointro,axiom,
! [M: mat_int] :
( ( incide6080938071136783841_1_int @ M )
=> ( incide8301514189696901506ix_int @ M ) ) ).
% zero_one_matrix_int.intro
thf(fact_65_zero__one__matrix__int_Oaxioms,axiom,
! [M: mat_int] :
( ( incide8301514189696901506ix_int @ M )
=> ( incide6080938071136783841_1_int @ M ) ) ).
% zero_one_matrix_int.axioms
thf(fact_66_zero__one__matrix__ring__1_Ointro,axiom,
! [M: mat_int] :
( ( incide4964164200581851450ix_int @ M )
=> ( incide6080938071136783841_1_int @ M ) ) ).
% zero_one_matrix_ring_1.intro
thf(fact_67_zero__one__matrix__ring__1_Oaxioms,axiom,
! [M: mat_int] :
( ( incide6080938071136783841_1_int @ M )
=> ( incide4964164200581851450ix_int @ M ) ) ).
% zero_one_matrix_ring_1.axioms
thf(fact_68_zero__one__matrix__int__def,axiom,
incide8301514189696901506ix_int = incide6080938071136783841_1_int ).
% zero_one_matrix_int_def
thf(fact_69_zero__one__matrix__ring__1__def,axiom,
incide6080938071136783841_1_int = incide4964164200581851450ix_int ).
% zero_one_matrix_ring_1_def
thf(fact_70_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_71_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_72_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_73_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_74_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: nat > complex,J2: complex > nat,T: set_nat,H: nat > int,G: complex > int] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_75_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: nat > complex,J2: complex > nat,T: set_nat,H: nat > real,G: complex > real] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_76_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: nat > complex,J2: complex > nat,T: set_nat,H: nat > nat,G: complex > nat] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_77_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: int > complex,J2: complex > int,T: set_int,H: int > real,G: complex > real] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_int @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S )
= ( groups8778361861064173332t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_78_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: int > complex,J2: complex > int,T: set_int,H: int > int,G: complex > int] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_int @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S )
= ( groups4538972089207619220nt_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_79_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: int > complex,J2: complex > int,T: set_int,H: int > nat,G: complex > nat] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_int @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S )
= ( groups4541462559716669496nt_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_80_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: complex > nat,J2: nat > complex,T: set_complex,H: complex > int,G: nat > int] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_complex @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups5690904116761175830ex_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_81_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,H: nat > int,G: nat > int] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_nat @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_82_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: int > nat,J2: nat > int,T: set_int,H: int > int,G: nat > int] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_int @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups4538972089207619220nt_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_83_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: complex > complex,J2: complex > complex,T: set_complex,H: complex > complex,G: complex > complex] :
( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( I2 @ ( J2 @ A2 ) )
= A2 ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( member_complex @ ( J2 @ A2 ) @ T ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( ( J2 @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T )
=> ( member_complex @ ( I2 @ B2 ) @ S ) )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ S )
=> ( ( H @ ( J2 @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_84_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > complex,A: set_complex,H: complex > nat,Gamma: nat > int,Phi: complex > int] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5690904116761175830ex_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_85_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > complex,A: set_complex,H: complex > nat,Gamma: nat > real,Phi: complex > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_86_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > complex,A: set_complex,H: complex > nat,Gamma: nat > nat,Phi: complex > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_87_sum_Oeq__general__inverses,axiom,
! [B: set_int,K: int > complex,A: set_complex,H: complex > int,Gamma: int > real,Phi: complex > real] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups8778361861064173332t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_88_sum_Oeq__general__inverses,axiom,
! [B: set_int,K: int > complex,A: set_complex,H: complex > int,Gamma: int > int,Phi: complex > int] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5690904116761175830ex_int @ Phi @ A )
= ( groups4538972089207619220nt_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_89_sum_Oeq__general__inverses,axiom,
! [B: set_int,K: int > complex,A: set_complex,H: complex > int,Gamma: int > nat,Phi: complex > nat] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups4541462559716669496nt_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_90_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > nat,A: set_nat,H: nat > complex,Gamma: complex > int,Phi: nat > int] :
( ! [Y: complex] :
( ( member_complex @ Y @ B )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_complex @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups5690904116761175830ex_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_91_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_92_sum_Oeq__general__inverses,axiom,
! [B: set_int,K: int > nat,A: set_nat,H: nat > int,Gamma: int > int,Phi: nat > int] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups4538972089207619220nt_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_93_sum_Oeq__general__inverses,axiom,
! [B: set_complex,K: complex > complex,A: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y: complex] :
( ( member_complex @ Y @ B )
=> ( ( member_complex @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_complex @ ( H @ X ) @ B )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_94_sum_Oeq__general,axiom,
! [B: set_nat,A: set_complex,H: complex > nat,Gamma: nat > int,Phi: complex > int] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5690904116761175830ex_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_95_sum_Oeq__general,axiom,
! [B: set_nat,A: set_complex,H: complex > nat,Gamma: nat > real,Phi: complex > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_96_sum_Oeq__general,axiom,
! [B: set_nat,A: set_complex,H: complex > nat,Gamma: nat > nat,Phi: complex > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_97_sum_Oeq__general,axiom,
! [B: set_int,A: set_complex,H: complex > int,Gamma: int > real,Phi: complex > real] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5808333547571424918x_real @ Phi @ A )
= ( groups8778361861064173332t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_98_sum_Oeq__general,axiom,
! [B: set_int,A: set_complex,H: complex > int,Gamma: int > int,Phi: complex > int] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5690904116761175830ex_int @ Phi @ A )
= ( groups4538972089207619220nt_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_99_sum_Oeq__general,axiom,
! [B: set_int,A: set_complex,H: complex > int,Gamma: int > nat,Phi: complex > nat] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5693394587270226106ex_nat @ Phi @ A )
= ( groups4541462559716669496nt_nat @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_100_sum_Oeq__general,axiom,
! [B: set_complex,A: set_nat,H: nat > complex,Gamma: complex > int,Phi: nat > int] :
( ! [Y: complex] :
( ( member_complex @ Y @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_complex @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups5690904116761175830ex_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_101_sum_Oeq__general,axiom,
! [B: set_nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_102_sum_Oeq__general,axiom,
! [B: set_int,A: set_nat,H: nat > int,Gamma: int > int,Phi: nat > int] :
( ! [Y: int] :
( ( member_int @ Y @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_int @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups4538972089207619220nt_int @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_103_sum_Oeq__general,axiom,
! [B: set_complex,A: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y: complex] :
( ( member_complex @ Y @ B )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ( member_complex @ ( H @ X ) @ B )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A )
= ( groups7754918857620584856omplex @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_104_index__row_I2_J,axiom,
! [A: mat_int,I2: nat] :
( ( dim_vec_int @ ( row_int @ A @ I2 ) )
= ( dim_col_int @ A ) ) ).
% index_row(2)
thf(fact_105_lift__mat__is__0__1,axiom,
incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ m ) ).
% lift_mat_is_0_1
thf(fact_106_dim__vec__mult__vec__mat,axiom,
! [V2: vec_int,A: mat_int] :
( ( dim_vec_int @ ( macaul1993098418559423895at_int @ V2 @ A ) )
= ( dim_col_int @ A ) ) ).
% dim_vec_mult_vec_mat
thf(fact_107_all__ones__mat__dim__col,axiom,
! [N: nat] :
( ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_col
thf(fact_108_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_109_lift__01__mat__simp_I2_J,axiom,
! [M: mat_int] :
( ( dim_col_int @ ( matrix323868623736973467nt_int @ M ) )
= ( dim_col_int @ M ) ) ).
% lift_01_mat_simp(2)
thf(fact_110_sum_Odelta_H,axiom,
! [S: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_111_sum_Odelta_H,axiom,
! [S: set_complex,A3: complex,B3: complex > int] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K2: complex] : ( if_int @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K2: complex] : ( if_int @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_112_sum_Odelta_H,axiom,
! [S: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_113_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_114_sum_Odelta_H,axiom,
! [S: set_complex,A3: complex,B3: complex > complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_115_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_116_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_117_sum_Odelta_H,axiom,
! [S: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A3 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_118_sum_Odelta_H,axiom,
! [S: set_int,A3: int,B3: int > complex] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A3 @ S )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_119_sum_Odelta_H,axiom,
! [S: set_nat,A3: nat,B3: nat > complex] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A3 = K2 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_120_sum_Odelta,axiom,
! [S: set_complex,A3: complex,B3: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_121_sum_Odelta,axiom,
! [S: set_complex,A3: complex,B3: complex > int] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K2: complex] : ( if_int @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K2: complex] : ( if_int @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_122_sum_Odelta,axiom,
! [S: set_complex,A3: complex,B3: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_123_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_124_sum_Odelta,axiom,
! [S: set_complex,A3: complex,B3: complex > complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A3 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_complex @ A3 @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_125_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_126_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_127_sum_Odelta,axiom,
! [S: set_int,A3: int,B3: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A3 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K2: int] : ( if_real @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_128_sum_Odelta,axiom,
! [S: set_int,A3: int,B3: int > complex] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A3 @ S )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_int @ A3 @ S )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_129_sum_Odelta,axiom,
! [S: set_nat,A3: nat,B3: nat > complex] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A3 @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member_nat @ A3 @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A3 ) @ ( B3 @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_130_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_131_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_132_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_133_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > complex,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_134_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_135_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_136_finite__atLeastLessThan,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).
% finite_atLeastLessThan
thf(fact_137_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_138_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_139_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_140_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_141_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups7754918857620584856omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_142_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_143_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_144_sum_Oinfinite,axiom,
! [A: set_int,G: int > real] :
( ~ ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_145_sum_Oinfinite,axiom,
! [A: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A )
=> ( ( groups3049146728041665814omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_146_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups2073611262835488442omplex @ G @ A )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_147_sum__eq__0__iff,axiom,
! [F2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_148_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_149_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: int] :
( ( member_int @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_150_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_151_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_152_sum__eq__Suc0__iff,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y4: complex] :
( ( member_complex @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_153_sum__eq__Suc0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_154_sum__eq__Suc0__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y4: int] :
( ( member_int @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_155_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
! [A3: complex,P: complex > $o] :
( ( member_complex @ A3 @ ( collect_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_161_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_162_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X: int] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_163_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X: complex] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_164_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_165_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_166_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_167_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_168_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_169_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_170_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ X @ Y )
=> ( P @ ( suc @ X ) @ ( suc @ Y ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_171_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_172_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_173_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_174_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_175_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_176_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_nat,G: complex > nat > int,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5690904116761175830ex_int
@ ^ [X4: complex] :
( groups3539618377306564664at_int @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups3539618377306564664at_int
@ ^ [Y4: nat] :
( groups5690904116761175830ex_int
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_177_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_nat,G: complex > nat > real,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5808333547571424918x_real
@ ^ [X4: complex] :
( groups6591440286371151544t_real @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups5808333547571424918x_real
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_178_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X4: complex] :
( groups3542108847815614940at_nat @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y4: nat] :
( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_179_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_int,G: complex > int > real,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_int @ B )
=> ( ( groups5808333547571424918x_real
@ ^ [X4: complex] :
( groups8778361861064173332t_real @ ( G @ X4 )
@ ( collect_int
@ ^ [Y4: int] :
( ( member_int @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups8778361861064173332t_real
@ ^ [Y4: int] :
( groups5808333547571424918x_real
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_180_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_int @ B )
=> ( ( groups5690904116761175830ex_int
@ ^ [X4: complex] :
( groups4538972089207619220nt_int @ ( G @ X4 )
@ ( collect_int
@ ^ [Y4: int] :
( ( member_int @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4538972089207619220nt_int
@ ^ [Y4: int] :
( groups5690904116761175830ex_int
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_181_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_int,G: complex > int > nat,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_int @ B )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X4: complex] :
( groups4541462559716669496nt_nat @ ( G @ X4 )
@ ( collect_int
@ ^ [Y4: int] :
( ( member_int @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4541462559716669496nt_nat
@ ^ [Y4: int] :
( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_182_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_complex,G: nat > complex > int,R: nat > complex > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( groups3539618377306564664at_int
@ ^ [X4: nat] :
( groups5690904116761175830ex_int @ ( G @ X4 )
@ ( collect_complex
@ ^ [Y4: complex] :
( ( member_complex @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups5690904116761175830ex_int
@ ^ [Y4: complex] :
( groups3539618377306564664at_int
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_183_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_nat,G: nat > nat > int,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( groups3539618377306564664at_int
@ ^ [X4: nat] :
( groups3539618377306564664at_int @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups3539618377306564664at_int
@ ^ [Y4: nat] :
( groups3539618377306564664at_int
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_184_sum_Oswap__restrict,axiom,
! [A: set_nat,B: set_int,G: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B )
=> ( ( groups3539618377306564664at_int
@ ^ [X4: nat] :
( groups4538972089207619220nt_int @ ( G @ X4 )
@ ( collect_int
@ ^ [Y4: int] :
( ( member_int @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups4538972089207619220nt_int
@ ^ [Y4: int] :
( groups3539618377306564664at_int
@ ^ [X4: nat] : ( G @ X4 @ Y4 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_185_sum_Oswap__restrict,axiom,
! [A: set_complex,B: set_complex,G: complex > complex > complex,R: complex > complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] :
( groups7754918857620584856omplex @ ( G @ X4 )
@ ( collect_complex
@ ^ [Y4: complex] :
( ( member_complex @ Y4 @ B )
& ( R @ X4 @ Y4 ) ) ) )
@ A )
= ( groups7754918857620584856omplex
@ ^ [Y4: complex] :
( groups7754918857620584856omplex
@ ^ [X4: complex] : ( G @ X4 @ Y4 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( R @ X4 @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_186_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_187_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_188_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_189_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > complex,M2: nat,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_190_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > int,G: nat > int] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ ( suc @ X ) @ A )
=> ( ( F @ ( suc @ X ) )
= ( G @ ( suc @ X ) ) ) )
=> ( ( groups3539618377306564664at_int @ F @ A )
= ( groups3539618377306564664at_int @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_191_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ ( suc @ X ) @ A )
=> ( ( F @ ( suc @ X ) )
= ( G @ ( suc @ X ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A )
= ( groups6591440286371151544t_real @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_192_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ ( suc @ X ) @ A )
=> ( ( F @ ( suc @ X ) )
= ( G @ ( suc @ X ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_193_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > complex,G: nat > complex] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ ( suc @ X ) @ A )
=> ( ( F @ ( suc @ X ) )
= ( G @ ( suc @ X ) ) ) )
=> ( ( groups2073611262835488442omplex @ F @ A )
= ( groups2073611262835488442omplex @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_194_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_195_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_196_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > int,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups5690904116761175830ex_int
@ ^ [X4: complex] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_197_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X4: complex] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_198_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X4: nat] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_199_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > complex,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups7754918857620584856omplex @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_complex )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_200_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_201_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_202_sum_Ointer__filter,axiom,
! [A: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X4: int] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_203_sum_Ointer__filter,axiom,
! [A: set_int,G: int > complex,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups3049146728041665814omplex @ G
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3049146728041665814omplex
@ ^ [X4: int] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_complex )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_204_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > complex,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups2073611262835488442omplex @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups2073611262835488442omplex
@ ^ [X4: nat] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_complex )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_205_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_206_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_207_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_208_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_209_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_210_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_211_finite__neq__0,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_212_finite__neq__0,axiom,
! [F: int > nat,G: int > nat,H: int > nat > nat > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: int] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_213_finite__neq__0,axiom,
! [F: complex > nat,G: complex > nat,H: complex > nat > nat > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: complex] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_214_finite__neq__0,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_215_finite__neq__0,axiom,
! [F: int > nat,G: int > nat,H: int > nat > nat > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: int] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_216_finite__neq__0,axiom,
! [F: complex > nat,G: complex > nat,H: complex > nat > nat > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: complex] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_217_finite__neq__0,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_218_finite__neq__0,axiom,
! [F: int > nat,G: int > nat,H: int > nat > nat > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: int] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_219_finite__neq__0,axiom,
! [F: complex > nat,G: complex > nat,H: complex > nat > nat > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: complex] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_220_finite__neq__0,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ X @ zero_zero_nat @ zero_zero_nat )
= zero_zero_real )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% finite_neq_0
thf(fact_221_finite__neq__0_H,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_222_finite__neq__0_H,axiom,
! [F: int > nat,G: int > nat,H: nat > nat > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_223_finite__neq__0_H,axiom,
! [F: complex > nat,G: complex > nat,H: nat > nat > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_nat )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_224_finite__neq__0_H,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_225_finite__neq__0_H,axiom,
! [F: int > nat,G: int > nat,H: nat > nat > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_226_finite__neq__0_H,axiom,
! [F: complex > nat,G: complex > nat,H: nat > nat > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_int )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_227_finite__neq__0_H,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_228_finite__neq__0_H,axiom,
! [F: int > nat,G: int > nat,H: nat > nat > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_229_finite__neq__0_H,axiom,
! [F: complex > nat,G: complex > nat,H: nat > nat > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_complex )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_230_finite__neq__0_H,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) )
=> ( ( ( H @ zero_zero_nat @ zero_zero_nat )
= zero_zero_real )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% finite_neq_0'
thf(fact_231_finite__neq__0__inv,axiom,
! [H: nat > nat > nat > nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat,Y: nat] :
( ( H @ X @ zero_zero_nat @ Y )
= Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_232_finite__neq__0__inv,axiom,
! [H: int > nat > nat > nat,F: int > nat,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: int,Y: nat] :
( ( H @ X @ zero_zero_nat @ Y )
= Y )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_233_finite__neq__0__inv,axiom,
! [H: complex > nat > nat > nat,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: complex,Y: nat] :
( ( H @ X @ zero_zero_nat @ Y )
= Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_234_finite__neq__0__inv,axiom,
! [H: nat > int > nat > nat,F: nat > int,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: nat,Y: nat] :
( ( H @ X @ zero_zero_int @ Y )
= Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_235_finite__neq__0__inv,axiom,
! [H: int > int > nat > nat,F: int > int,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: int,Y: nat] :
( ( H @ X @ zero_zero_int @ Y )
= Y )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_236_finite__neq__0__inv,axiom,
! [H: complex > int > nat > nat,F: complex > int,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: complex,Y: nat] :
( ( H @ X @ zero_zero_int @ Y )
= Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_237_finite__neq__0__inv,axiom,
! [H: nat > complex > nat > nat,F: nat > complex,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: nat,Y: nat] :
( ( H @ X @ zero_zero_complex @ Y )
= Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_238_finite__neq__0__inv,axiom,
! [H: int > complex > nat > nat,F: int > complex,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: int,Y: nat] :
( ( H @ X @ zero_zero_complex @ Y )
= Y )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_239_finite__neq__0__inv,axiom,
! [H: complex > complex > nat > nat,F: complex > complex,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: complex,Y: nat] :
( ( H @ X @ zero_zero_complex @ Y )
= Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_240_finite__neq__0__inv,axiom,
! [H: nat > real > nat > nat,F: nat > real,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ X4 @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_real ) ) )
=> ( ! [X: nat,Y: nat] :
( ( H @ X @ zero_zero_real @ Y )
= Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv
thf(fact_241_finite__neq__0__inv_H,axiom,
! [H: nat > nat > nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_nat @ X )
= X )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_242_finite__neq__0__inv_H,axiom,
! [H: nat > nat > nat,F: int > nat,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_nat @ X )
= X )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_243_finite__neq__0__inv_H,axiom,
! [H: nat > nat > nat,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_nat ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_nat @ X )
= X )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_244_finite__neq__0__inv_H,axiom,
! [H: int > nat > nat,F: nat > int,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_int @ X )
= X )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_245_finite__neq__0__inv_H,axiom,
! [H: int > nat > nat,F: int > int,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_int @ X )
= X )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_246_finite__neq__0__inv_H,axiom,
! [H: int > nat > nat,F: complex > int,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_int ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_int @ X )
= X )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_247_finite__neq__0__inv_H,axiom,
! [H: complex > nat > nat,F: nat > complex,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_complex @ X )
= X )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_248_finite__neq__0__inv_H,axiom,
! [H: complex > nat > nat,F: int > complex,G: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_complex @ X )
= X )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_249_finite__neq__0__inv_H,axiom,
! [H: complex > nat > nat,F: complex > complex,G: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( F @ X4 )
!= zero_zero_complex ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_complex @ X )
= X )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_250_finite__neq__0__inv_H,axiom,
! [H: real > nat > nat,F: nat > real,G: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( H @ ( F @ X4 ) @ ( G @ X4 ) )
!= zero_zero_nat ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( F @ X4 )
!= zero_zero_real ) ) )
=> ( ! [X: nat] :
( ( H @ zero_zero_real @ X )
= X )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
!= zero_zero_nat ) ) ) ) ) ) ).
% finite_neq_0_inv'
thf(fact_251_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_252_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_253_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_12: nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_254_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_12: int] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_255_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_12: complex] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_256_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_257_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_258_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_complex,R: nat > complex > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_259_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_260_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_261_pigeonhole__infinite__rel,axiom,
! [A: set_int,B: set_complex,R: int > complex > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_262_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_nat,R: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_263_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_int,R: complex > int > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_int @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_264_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_complex,R: complex > complex > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_265_index__update__vec2,axiom,
! [I3: nat,I2: nat,V2: vec_int,A3: int] :
( ( I3 != I2 )
=> ( ( vec_index_int @ ( update_vec_int @ V2 @ I2 @ A3 ) @ I3 )
= ( vec_index_int @ V2 @ I3 ) ) ) ).
% index_update_vec2
thf(fact_266_mat__of__row__dim_I2_J,axiom,
! [Y3: vec_int] :
( ( dim_col_int @ ( mat_of_row_int @ Y3 ) )
= ( dim_vec_int @ Y3 ) ) ).
% mat_of_row_dim(2)
thf(fact_267_row__mat__of__row,axiom,
! [Y3: vec_int] :
( ( row_int @ ( mat_of_row_int @ Y3 ) @ zero_zero_nat )
= Y3 ) ).
% row_mat_of_row
thf(fact_268_dim__update__mat_I2_J,axiom,
! [A: mat_int,Ij: product_prod_nat_nat,A3: int] :
( ( dim_col_int @ ( update_mat_int @ A @ Ij @ A3 ) )
= ( dim_col_int @ A ) ) ).
% dim_update_mat(2)
thf(fact_269_sum_Onested__swap,axiom,
! [A3: nat > nat > int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups3539618377306564664at_int @ ( A3 @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_270_sum_Onested__swap,axiom,
! [A3: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( A3 @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_271_sum_Onested__swap,axiom,
! [A3: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( A3 @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_272_sum_Onested__swap,axiom,
! [A3: nat > nat > complex,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( groups2073611262835488442omplex @ ( A3 @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [J: nat] :
( groups2073611262835488442omplex
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_273_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_274_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_275_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_276_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_277_add__left__cancel,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_278_add__left__cancel,axiom,
! [A3: int,B3: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_279_add__left__cancel,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_280_add__right__cancel,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_281_add__right__cancel,axiom,
! [B3: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_282_add__right__cancel,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_283_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_284_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_285_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_286_index__add__mat_I3_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_col_int @ ( plus_plus_mat_int @ A @ B ) )
= ( dim_col_int @ B ) ) ).
% index_add_mat(3)
thf(fact_287_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_288_add_Oright__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.right_neutral
thf(fact_289_add_Oright__neutral,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% add.right_neutral
thf(fact_290_add_Oright__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.right_neutral
thf(fact_291_double__zero__sym,axiom,
! [A3: int] :
( ( zero_zero_int
= ( plus_plus_int @ A3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_292_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_293_add__cancel__left__left,axiom,
! [B3: nat,A3: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_294_add__cancel__left__left,axiom,
! [B3: int,A3: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_295_add__cancel__left__left,axiom,
! [B3: complex,A3: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_296_add__cancel__left__left,axiom,
! [B3: real,A3: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_297_add__cancel__left__right,axiom,
! [A3: nat,B3: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_298_add__cancel__left__right,axiom,
! [A3: int,B3: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_299_add__cancel__left__right,axiom,
! [A3: complex,B3: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_300_add__cancel__left__right,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_301_add__cancel__right__left,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_302_add__cancel__right__left,axiom,
! [A3: int,B3: int] :
( ( A3
= ( plus_plus_int @ B3 @ A3 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_303_add__cancel__right__left,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( plus_plus_complex @ B3 @ A3 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_304_add__cancel__right__left,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ B3 @ A3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_305_add__cancel__right__right,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_306_add__cancel__right__right,axiom,
! [A3: int,B3: int] :
( ( A3
= ( plus_plus_int @ A3 @ B3 ) )
= ( B3 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_307_add__cancel__right__right,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( plus_plus_complex @ A3 @ B3 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_308_add__cancel__right__right,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ A3 @ B3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_309_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_310_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_311_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_312_add__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add_0
thf(fact_313_add__0,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% add_0
thf(fact_314_add__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add_0
thf(fact_315_finite__atLeastAtMost,axiom,
! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% finite_atLeastAtMost
thf(fact_316_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_317_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_318_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_319_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( I2 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_320_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J2: int,K: int,L2: int] :
( ( ( I2 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_321_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J2: real,K: real,L2: real] :
( ( ( I2 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_322_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_323_group__cancel_Oadd1,axiom,
! [A: int,K: int,A3: int,B3: int] :
( ( A
= ( plus_plus_int @ K @ A3 ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_324_group__cancel_Oadd1,axiom,
! [A: real,K: real,A3: real,B3: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_325_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B3: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_326_group__cancel_Oadd2,axiom,
! [B: int,K: int,B3: int,A3: int] :
( ( B
= ( plus_plus_int @ K @ B3 ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_327_group__cancel_Oadd2,axiom,
! [B: real,K: real,B3: real,A3: real] :
( ( B
= ( plus_plus_real @ K @ B3 ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_328_add_Oassoc,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add.assoc
thf(fact_329_add_Oassoc,axiom,
! [A3: int,B3: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% add.assoc
thf(fact_330_add_Oassoc,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% add.assoc
thf(fact_331_add_Oleft__cancel,axiom,
! [A3: int,B3: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add.left_cancel
thf(fact_332_add_Oleft__cancel,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add.left_cancel
thf(fact_333_add_Oright__cancel,axiom,
! [B3: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add.right_cancel
thf(fact_334_add_Oright__cancel,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add.right_cancel
thf(fact_335_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_336_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_337_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_338_add_Oleft__commute,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add.left_commute
thf(fact_339_add_Oleft__commute,axiom,
! [B3: int,A3: int,C2: int] :
( ( plus_plus_int @ B3 @ ( plus_plus_int @ A3 @ C2 ) )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% add.left_commute
thf(fact_340_add_Oleft__commute,axiom,
! [B3: real,A3: real,C2: real] :
( ( plus_plus_real @ B3 @ ( plus_plus_real @ A3 @ C2 ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% add.left_commute
thf(fact_341_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_342_add__left__imp__eq,axiom,
! [A3: int,B3: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B3 )
= ( plus_plus_int @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_343_add__left__imp__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_344_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_345_add__right__imp__eq,axiom,
! [B3: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B3 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_346_add__right__imp__eq,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_347_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M2: nat,K: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_348_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M2: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_349_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_350_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > complex,M2: nat,K: nat,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_351_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_352_comm__monoid__add__class_Oadd__0,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_353_comm__monoid__add__class_Oadd__0,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_354_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_355_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_356_add_Ocomm__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% add.comm_neutral
thf(fact_357_add_Ocomm__neutral,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% add.comm_neutral
thf(fact_358_add_Ocomm__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% add.comm_neutral
thf(fact_359_add_Ogroup__left__neutral,axiom,
! [A3: int] :
( ( plus_plus_int @ zero_zero_int @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_360_add_Ogroup__left__neutral,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_361_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_362_plus__eq__zero,axiom,
! [S2: nat,T2: nat] :
( ( ( plus_plus_nat @ S2 @ T2 )
= zero_zero_nat )
=> ( S2 = zero_zero_nat ) ) ).
% plus_eq_zero
thf(fact_363_plus__eq__zero__2,axiom,
! [S2: nat,T2: nat] :
( ( ( plus_plus_nat @ S2 @ T2 )
= zero_zero_nat )
=> ( T2 = zero_zero_nat ) ) ).
% plus_eq_zero_2
thf(fact_364_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_365_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_366_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_367_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_368_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_369_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_370_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_371_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_372_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_373_sum_Odistrib,axiom,
! [G: nat > int,H: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X4: nat] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A ) @ ( groups3539618377306564664at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_374_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A ) @ ( groups7754918857620584856omplex @ H @ A ) ) ) ).
% sum.distrib
thf(fact_375_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A ) @ ( groups6591440286371151544t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_376_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_377_sum_Odistrib,axiom,
! [G: int > real,H: int > real,A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [X4: int] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ A ) @ ( groups8778361861064173332t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_378_sum_Odistrib,axiom,
! [G: int > complex,H: int > complex,A: set_int] :
( ( groups3049146728041665814omplex
@ ^ [X4: int] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_complex @ ( groups3049146728041665814omplex @ G @ A ) @ ( groups3049146728041665814omplex @ H @ A ) ) ) ).
% sum.distrib
thf(fact_379_sum_Odistrib,axiom,
! [G: nat > complex,H: nat > complex,A: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [X4: nat] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ A ) @ ( groups2073611262835488442omplex @ H @ A ) ) ) ).
% sum.distrib
thf(fact_380_sum_Odistrib,axiom,
! [G: int > int,H: int > int,A: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A ) @ ( groups4538972089207619220nt_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_381_sum_Odistrib,axiom,
! [G: int > nat,H: int > nat,A: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [X4: int] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A ) @ ( groups4541462559716669496nt_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_382_double__sum__split__case2,axiom,
! [G: complex > complex > nat,A: set_complex] :
( ( groups5693394587270226106ex_nat
@ ^ [I: complex] : ( groups5693394587270226106ex_nat @ ( G @ I ) @ A )
@ A )
= ( plus_plus_nat
@ ( groups5693394587270226106ex_nat
@ ^ [I: complex] : ( G @ I @ I )
@ A )
@ ( groups5693394587270226106ex_nat
@ ^ [I: complex] :
( groups5693394587270226106ex_nat @ ( G @ I )
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_383_double__sum__split__case2,axiom,
! [G: complex > complex > int,A: set_complex] :
( ( groups5690904116761175830ex_int
@ ^ [I: complex] : ( groups5690904116761175830ex_int @ ( G @ I ) @ A )
@ A )
= ( plus_plus_int
@ ( groups5690904116761175830ex_int
@ ^ [I: complex] : ( G @ I @ I )
@ A )
@ ( groups5690904116761175830ex_int
@ ^ [I: complex] :
( groups5690904116761175830ex_int @ ( G @ I )
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_384_double__sum__split__case2,axiom,
! [G: complex > complex > real,A: set_complex] :
( ( groups5808333547571424918x_real
@ ^ [I: complex] : ( groups5808333547571424918x_real @ ( G @ I ) @ A )
@ A )
= ( plus_plus_real
@ ( groups5808333547571424918x_real
@ ^ [I: complex] : ( G @ I @ I )
@ A )
@ ( groups5808333547571424918x_real
@ ^ [I: complex] :
( groups5808333547571424918x_real @ ( G @ I )
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_385_double__sum__split__case2,axiom,
! [G: nat > nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups3539618377306564664at_int @ ( G @ I ) @ A )
@ A )
= ( plus_plus_int
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ I @ I )
@ A )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups3539618377306564664at_int @ ( G @ I )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_386_double__sum__split__case2,axiom,
! [G: complex > complex > complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ A )
@ A )
= ( plus_plus_complex
@ ( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ I )
@ A )
@ ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex @ ( G @ I )
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_387_double__sum__split__case2,axiom,
! [G: nat > nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ A )
@ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ I )
@ A )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real @ ( G @ I )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_388_double__sum__split__case2,axiom,
! [G: nat > nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ A )
@ A )
= ( plus_plus_nat
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ I )
@ A )
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat @ ( G @ I )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_389_double__sum__split__case2,axiom,
! [G: int > int > real,A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [I: int] : ( groups8778361861064173332t_real @ ( G @ I ) @ A )
@ A )
= ( plus_plus_real
@ ( groups8778361861064173332t_real
@ ^ [I: int] : ( G @ I @ I )
@ A )
@ ( groups8778361861064173332t_real
@ ^ [I: int] :
( groups8778361861064173332t_real @ ( G @ I )
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_390_double__sum__split__case2,axiom,
! [G: int > int > complex,A: set_int] :
( ( groups3049146728041665814omplex
@ ^ [I: int] : ( groups3049146728041665814omplex @ ( G @ I ) @ A )
@ A )
= ( plus_plus_complex
@ ( groups3049146728041665814omplex
@ ^ [I: int] : ( G @ I @ I )
@ A )
@ ( groups3049146728041665814omplex
@ ^ [I: int] :
( groups3049146728041665814omplex @ ( G @ I )
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_391_double__sum__split__case2,axiom,
! [G: nat > nat > complex,A: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I: nat] : ( groups2073611262835488442omplex @ ( G @ I ) @ A )
@ A )
= ( plus_plus_complex
@ ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ I @ I )
@ A )
@ ( groups2073611262835488442omplex
@ ^ [I: nat] :
( groups2073611262835488442omplex @ ( G @ I )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( A4 != I ) ) ) )
@ A ) ) ) ).
% double_sum_split_case2
thf(fact_392_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_393_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_394_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_395_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_396_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_397_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_398_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > complex,M2: nat,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_399_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X2: nat > nat,Y3: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( plus_plus_nat @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_400_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X2: int > nat,Y3: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( plus_plus_nat @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_401_sum_Ofinite__Collect__op,axiom,
! [I4: set_complex,X2: complex > nat,Y3: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( plus_plus_nat @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_402_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X2: nat > int,Y3: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( plus_plus_int @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_403_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X2: int > int,Y3: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( plus_plus_int @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_404_sum_Ofinite__Collect__op,axiom,
! [I4: set_complex,X2: complex > int,Y3: complex > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_int ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( plus_plus_int @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_405_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X2: nat > complex,Y3: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( plus_plus_complex @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_406_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X2: int > complex,Y3: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I4 )
& ( ( plus_plus_complex @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_407_sum_Ofinite__Collect__op,axiom,
! [I4: set_complex,X2: complex > complex,Y3: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I4 )
& ( ( plus_plus_complex @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_408_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X2: nat > real,Y3: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( X2 @ I )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( Y3 @ I )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I4 )
& ( ( plus_plus_real @ ( X2 @ I ) @ ( Y3 @ I ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_409_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > int,M2: nat,K: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_410_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > real,M2: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_411_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_412_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > complex,M2: nat,K: nat,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_413_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_complex,H: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_414_sum_Orelated,axiom,
! [R: int > int > $o,S: set_complex,H: complex > int,G: complex > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y22: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups5690904116761175830ex_int @ H @ S ) @ ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_415_sum_Orelated,axiom,
! [R: real > real > $o,S: set_complex,H: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y22: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_416_sum_Orelated,axiom,
! [R: int > int > $o,S: set_nat,H: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y22: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_417_sum_Orelated,axiom,
! [R: complex > complex > $o,S: set_complex,H: complex > complex,G: complex > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X: complex] :
( ( member_complex @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups7754918857620584856omplex @ H @ S ) @ ( groups7754918857620584856omplex @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_418_sum_Orelated,axiom,
! [R: real > real > $o,S: set_nat,H: nat > real,G: nat > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y22: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups6591440286371151544t_real @ H @ S ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_419_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_nat,H: nat > nat,G: nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3542108847815614940at_nat @ H @ S ) @ ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_420_sum_Orelated,axiom,
! [R: real > real > $o,S: set_int,H: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y22: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H @ S ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_421_sum_Orelated,axiom,
! [R: complex > complex > $o,S: set_int,H: int > complex,G: int > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3049146728041665814omplex @ H @ S ) @ ( groups3049146728041665814omplex @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_422_sum_Orelated,axiom,
! [R: complex > complex > $o,S: set_nat,H: nat > complex,G: nat > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups2073611262835488442omplex @ H @ S ) @ ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_423_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_424_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_425_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_426_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_427_double__eq__0__iff,axiom,
! [A3: int] :
( ( ( plus_plus_int @ A3 @ A3 )
= zero_zero_int )
= ( A3 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_428_double__eq__0__iff,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_429_fun__sum__commute,axiom,
! [F: nat > complex,G: complex > nat,A: set_complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5693394587270226106ex_nat @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_430_fun__sum__commute,axiom,
! [F: int > complex,G: complex > int,A: set_complex] :
( ( ( F @ zero_zero_int )
= zero_zero_complex )
=> ( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5690904116761175830ex_int @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_431_fun__sum__commute,axiom,
! [F: real > complex,G: complex > real,A: set_complex] :
( ( ( F @ zero_zero_real )
= zero_zero_complex )
=> ( ! [X: real,Y: real] :
( ( F @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5808333547571424918x_real @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_432_fun__sum__commute,axiom,
! [F: int > int,G: nat > int,A: set_nat] :
( ( ( F @ zero_zero_int )
= zero_zero_int )
=> ( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups3539618377306564664at_int
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_433_fun__sum__commute,axiom,
! [F: int > real,G: nat > int,A: set_nat] :
( ( ( F @ zero_zero_int )
= zero_zero_real )
=> ( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups6591440286371151544t_real
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_434_fun__sum__commute,axiom,
! [F: int > nat,G: nat > int,A: set_nat] :
( ( ( F @ zero_zero_int )
= zero_zero_nat )
=> ( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_435_fun__sum__commute,axiom,
! [F: int > complex,G: nat > int,A: set_nat] :
( ( ( F @ zero_zero_int )
= zero_zero_complex )
=> ( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups2073611262835488442omplex
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_436_fun__sum__commute,axiom,
! [F: complex > nat,G: complex > complex,A: set_complex] :
( ( ( F @ zero_zero_complex )
= zero_zero_nat )
=> ( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5693394587270226106ex_nat
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_437_fun__sum__commute,axiom,
! [F: complex > int,G: complex > complex,A: set_complex] :
( ( ( F @ zero_zero_complex )
= zero_zero_int )
=> ( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5690904116761175830ex_int
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_438_fun__sum__commute,axiom,
! [F: complex > real,G: complex > complex,A: set_complex] :
( ( ( F @ zero_zero_complex )
= zero_zero_real )
=> ( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5808333547571424918x_real
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ) ).
% fun_sum_commute
thf(fact_439_bound__nat__induct,axiom,
! [N: nat,L2: nat,U: nat,P: nat > $o] :
( ( member_nat @ N @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
=> ( ( P @ L2 )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( member_nat @ N2 @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% bound_nat_induct
thf(fact_440_fun__sum__commute__canc,axiom,
! [F: nat > complex,G: complex > nat,A: set_complex] :
( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5693394587270226106ex_nat @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_441_fun__sum__commute__canc,axiom,
! [F: int > complex,G: complex > int,A: set_complex] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5690904116761175830ex_int @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_442_fun__sum__commute__canc,axiom,
! [F: real > complex,G: complex > real,A: set_complex] :
( ! [X: real,Y: real] :
( ( F @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups5808333547571424918x_real @ G @ A ) )
= ( groups7754918857620584856omplex
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_443_fun__sum__commute__canc,axiom,
! [F: int > int,G: nat > int,A: set_nat] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups3539618377306564664at_int
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_444_fun__sum__commute__canc,axiom,
! [F: int > real,G: nat > int,A: set_nat] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups6591440286371151544t_real
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_445_fun__sum__commute__canc,axiom,
! [F: int > nat,G: nat > int,A: set_nat] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_446_fun__sum__commute__canc,axiom,
! [F: int > complex,G: nat > int,A: set_nat] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups3539618377306564664at_int @ G @ A ) )
= ( groups2073611262835488442omplex
@ ^ [A4: nat] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_447_fun__sum__commute__canc,axiom,
! [F: complex > nat,G: complex > complex,A: set_complex] :
( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5693394587270226106ex_nat
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_448_fun__sum__commute__canc,axiom,
! [F: complex > int,G: complex > complex,A: set_complex] :
( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5690904116761175830ex_int
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_449_fun__sum__commute__canc,axiom,
! [F: complex > real,G: complex > complex,A: set_complex] :
( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ ( groups7754918857620584856omplex @ G @ A ) )
= ( groups5808333547571424918x_real
@ ^ [A4: complex] : ( F @ ( G @ A4 ) )
@ A ) ) ) ).
% fun_sum_commute_canc
thf(fact_450_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A3: nat,B3: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2581907887559384638at_int
@ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_451_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A3: nat,B3: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_452_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_453_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A3: nat,B3: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
= ( set_fo1517530859248394432omplex
@ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
@ A3
@ B3
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_454_add__0__iff,axiom,
! [B3: nat,A3: nat] :
( ( B3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_455_add__0__iff,axiom,
! [B3: int,A3: int] :
( ( B3
= ( plus_plus_int @ B3 @ A3 ) )
= ( A3 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_456_add__0__iff,axiom,
! [B3: complex,A3: complex] :
( ( B3
= ( plus_plus_complex @ B3 @ A3 ) )
= ( A3 = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_457_add__0__iff,axiom,
! [B3: real,A3: real] :
( ( B3
= ( plus_plus_real @ B3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_458_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_459_verit__sum__simplify,axiom,
! [A3: int] :
( ( plus_plus_int @ A3 @ zero_zero_int )
= A3 ) ).
% verit_sum_simplify
thf(fact_460_verit__sum__simplify,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% verit_sum_simplify
thf(fact_461_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_462_additive__implies__homogenous,axiom,
! [F: nat > nat] :
( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_463_additive__implies__homogenous,axiom,
! [F: nat > int] :
( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_464_additive__implies__homogenous,axiom,
! [F: nat > complex] :
( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_complex ) ) ).
% additive_implies_homogenous
thf(fact_465_additive__implies__homogenous,axiom,
! [F: nat > real] :
( ! [X: nat,Y: nat] :
( ( F @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_nat )
= zero_zero_real ) ) ).
% additive_implies_homogenous
thf(fact_466_additive__implies__homogenous,axiom,
! [F: int > nat] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_467_additive__implies__homogenous,axiom,
! [F: int > int] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_468_additive__implies__homogenous,axiom,
! [F: int > complex] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_complex @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_complex ) ) ).
% additive_implies_homogenous
thf(fact_469_additive__implies__homogenous,axiom,
! [F: int > real] :
( ! [X: int,Y: int] :
( ( F @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_int )
= zero_zero_real ) ) ).
% additive_implies_homogenous
thf(fact_470_additive__implies__homogenous,axiom,
! [F: complex > nat] :
( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_complex )
= zero_zero_nat ) ) ).
% additive_implies_homogenous
thf(fact_471_additive__implies__homogenous,axiom,
! [F: complex > int] :
( ! [X: complex,Y: complex] :
( ( F @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ( F @ zero_zero_complex )
= zero_zero_int ) ) ).
% additive_implies_homogenous
thf(fact_472_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_473_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_474_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_475_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_476_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_477_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_478_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_479_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_480_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_481_add__less__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_482_add__less__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_483_add__less__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_484_add__less__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_485_add__less__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_486_add__less__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_487_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_488_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_489_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_490_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_491_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_492_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_493_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_494_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_495_add__less__same__cancel1,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B3 @ A3 ) @ B3 )
= ( ord_less_complex @ A3 @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_496_add__less__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_497_add__less__same__cancel1,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_498_add__less__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_499_add__less__same__cancel2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A3 @ B3 ) @ B3 )
= ( ord_less_complex @ A3 @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_500_add__less__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_501_add__less__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_502_add__less__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_503_less__add__same__cancel1,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ ( plus_plus_complex @ A3 @ B3 ) )
= ( ord_less_complex @ zero_zero_complex @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_504_less__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_505_less__add__same__cancel1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_506_less__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_507_less__add__same__cancel2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ ( plus_plus_complex @ B3 @ A3 ) )
= ( ord_less_complex @ zero_zero_complex @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_508_less__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_509_less__add__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_510_less__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_511_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_512_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_513_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_514_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_real @ zero_zero_real @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_515_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_516_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_517_infinite__Icc__iff,axiom,
! [A3: real,B3: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% infinite_Icc_iff
thf(fact_518_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_519_infinite__Ico__iff,axiom,
! [A3: real,B3: real] :
( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A3 @ B3 ) ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% infinite_Ico_iff
thf(fact_520_eq__vecI,axiom,
! [W: vec_int,V2: vec_int] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( dim_vec_int @ W ) )
=> ( ( vec_index_int @ V2 @ I5 )
= ( vec_index_int @ W @ I5 ) ) )
=> ( ( ( dim_vec_int @ V2 )
= ( dim_vec_int @ W ) )
=> ( V2 = W ) ) ) ).
% eq_vecI
thf(fact_521_index__update__vec1,axiom,
! [I2: nat,V2: vec_int,A3: int] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ V2 ) )
=> ( ( vec_index_int @ ( update_vec_int @ V2 @ I2 @ A3 ) @ I2 )
= A3 ) ) ).
% index_update_vec1
thf(fact_522_index__add__vec_I1_J,axiom,
! [I2: nat,V_2: vec_nat,V_1: vec_nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_nat @ V_2 ) )
=> ( ( vec_index_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ I2 )
= ( plus_plus_nat @ ( vec_index_nat @ V_1 @ I2 ) @ ( vec_index_nat @ V_2 @ I2 ) ) ) ) ).
% index_add_vec(1)
thf(fact_523_index__add__vec_I1_J,axiom,
! [I2: nat,V_2: vec_int,V_1: vec_int] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ V_2 ) )
=> ( ( vec_index_int @ ( plus_plus_vec_int @ V_1 @ V_2 ) @ I2 )
= ( plus_plus_int @ ( vec_index_int @ V_1 @ I2 ) @ ( vec_index_int @ V_2 @ I2 ) ) ) ) ).
% index_add_vec(1)
thf(fact_524_index__add__vec_I1_J,axiom,
! [I2: nat,V_2: vec_real,V_1: vec_real] :
( ( ord_less_nat @ I2 @ ( dim_vec_real @ V_2 ) )
=> ( ( vec_index_real @ ( plus_plus_vec_real @ V_1 @ V_2 ) @ I2 )
= ( plus_plus_real @ ( vec_index_real @ V_1 @ I2 ) @ ( vec_index_real @ V_2 @ I2 ) ) ) ) ).
% index_add_vec(1)
thf(fact_525_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_526_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_527_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_528_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_529_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_530_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_531_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_532_less__add__eq__less,axiom,
! [K: nat,L2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_533_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_534_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_535_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_536_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_537_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_538_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_539_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_540_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_541_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_542_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_543_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_544_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_545_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_546_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_547_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_548_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_549_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_550_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_551_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_552_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_553_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_554_add__less__imp__less__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_555_add__less__imp__less__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_556_add__less__imp__less__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_557_add__less__imp__less__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_558_add__less__imp__less__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_559_add__less__imp__less__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_560_add__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_561_add__strict__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_562_add__strict__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_563_add__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_564_add__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_565_add__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_566_add__strict__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_567_add__strict__mono,axiom,
! [A3: int,B3: int,C2: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_568_add__strict__mono,axiom,
! [A3: real,B3: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_569_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_570_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J2: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_571_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J2: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_572_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_573_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J2: int,K: int,L2: int] :
( ( ( I2 = J2 )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_574_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J2: real,K: real,L2: real] :
( ( ( I2 = J2 )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_575_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_576_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J2: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J2 )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_577_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J2: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_578_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_579_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_580_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_581_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_582_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_583_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_584_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_585_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_586_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_587_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_588_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_589_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_590_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_591_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_592_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_593_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_594_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_595_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_596_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_597_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_598_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_599_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_600_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
=> ( ! [I5: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I5 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I5 @ K3 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_601_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I5: nat] :
( ( J2
= ( suc @ I5 ) )
=> ( P @ I5 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ J2 )
=> ( ( P @ ( suc @ I5 ) )
=> ( P @ I5 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_602_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_603_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_604_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_605_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_606_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M6 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_607_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_608_atLeastLessThan__eq__iff,axiom,
! [A3: real,B3: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ( set_or66887138388493659n_real @ A3 @ B3 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
= ( ( A3 = C2 )
& ( B3 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_609_atLeastLessThan__eq__iff,axiom,
! [A3: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A3 @ B3 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
= ( ( A3 = C2 )
& ( B3 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_610_atLeastLessThan__eq__iff,axiom,
! [A3: int,B3: int,C2: int,D: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ( set_or4662586982721622107an_int @ A3 @ B3 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
= ( ( A3 = C2 )
& ( B3 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_611_Ico__eq__Ico,axiom,
! [L2: real,H: real,L3: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L2 @ H )
= ( set_or66887138388493659n_real @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L2 @ H )
& ~ ( ord_less_real @ L3 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_612_Ico__eq__Ico,axiom,
! [L2: nat,H: nat,L3: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L2 @ H )
= ( set_or4665077453230672383an_nat @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L2 @ H )
& ~ ( ord_less_nat @ L3 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_613_Ico__eq__Ico,axiom,
! [L2: int,H: int,L3: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L2 @ H )
= ( set_or4662586982721622107an_int @ L3 @ H2 ) )
= ( ( ( L2 = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L2 @ H )
& ~ ( ord_less_int @ L3 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_614_atLeastLessThan__inj_I1_J,axiom,
! [A3: real,B3: real,C2: real,D: real] :
( ( ( set_or66887138388493659n_real @ A3 @ B3 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
=> ( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( A3 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_615_atLeastLessThan__inj_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A3 @ B3 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( A3 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_616_atLeastLessThan__inj_I1_J,axiom,
! [A3: int,B3: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A3 @ B3 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( A3 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_617_atLeastLessThan__inj_I2_J,axiom,
! [A3: real,B3: real,C2: real,D: real] :
( ( ( set_or66887138388493659n_real @ A3 @ B3 )
= ( set_or66887138388493659n_real @ C2 @ D ) )
=> ( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( B3 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_618_atLeastLessThan__inj_I2_J,axiom,
! [A3: nat,B3: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A3 @ B3 )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( B3 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_619_atLeastLessThan__inj_I2_J,axiom,
! [A3: int,B3: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A3 @ B3 )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( B3 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_620_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_621_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_622_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_623_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% all_Suc_conv
thf(fact_624_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_625_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_626_pos__add__strict,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_complex @ B3 @ C2 )
=> ( ord_less_complex @ B3 @ ( plus_plus_complex @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_627_pos__add__strict,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_628_pos__add__strict,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_629_pos__add__strict,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_630_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_631_add__pos__pos,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_complex @ zero_zero_complex @ B3 )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_632_add__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_633_add__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_634_add__pos__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_635_add__neg__neg,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_complex @ B3 @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_neg_neg
thf(fact_636_add__neg__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_637_add__neg__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_638_add__neg__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_639_infinite__Icc,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) ) ).
% infinite_Icc
thf(fact_640_infinite__Ico,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A3 @ B3 ) ) ) ).
% infinite_Ico
thf(fact_641_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_642_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_643_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_644_vec__eq__iff,axiom,
( ( ^ [Y5: vec_int,Z: vec_int] : ( Y5 = Z ) )
= ( ^ [X4: vec_int,Y4: vec_int] :
( ( ( dim_vec_int @ X4 )
= ( dim_vec_int @ Y4 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_int @ Y4 ) )
=> ( ( vec_index_int @ X4 @ I )
= ( vec_index_int @ Y4 @ I ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_645_infinite__imp__elem,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ? [X: nat] : ( member_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_646_infinite__imp__elem,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ? [X: int] : ( member_int @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_647_infinite__imp__elem,axiom,
! [A: set_complex] :
( ~ ( finite3207457112153483333omplex @ A )
=> ? [X: complex] : ( member_complex @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_648_sum__SucD,axiom,
! [F: nat > nat,A: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ N ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ).
% sum_SucD
thf(fact_649_sum__SucD,axiom,
! [F: int > nat,A: set_int,N: nat] :
( ( ( groups4541462559716669496nt_nat @ F @ A )
= ( suc @ N ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ).
% sum_SucD
thf(fact_650_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_651_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_652_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_653_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > complex] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_complex @ ( G @ M2 ) @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_654_sum_Ohead__if,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_655_sum_Ohead__if,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_656_sum_Ohead__if,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_657_sum_Ohead__if,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_658_add__mset__in__multiset,axiom,
! [M: nat > nat,A3: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A3 ) @ ( suc @ ( M @ X4 ) ) @ ( M @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_659_add__mset__in__multiset,axiom,
! [M: int > nat,A3: int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A3 ) @ ( suc @ ( M @ X4 ) ) @ ( M @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_660_add__mset__in__multiset,axiom,
! [M: complex > nat,A3: complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A3 ) @ ( suc @ ( M @ X4 ) ) @ ( M @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_661_in__map__col__valid__index,axiom,
! [I2: nat,C2: vec_int] :
( ( member_nat @ I2 @ ( incide3973235006681262014ck_int @ C2 ) )
=> ( ord_less_nat @ I2 @ ( dim_vec_int @ C2 ) ) ) ).
% in_map_col_valid_index
thf(fact_662_filter__preserves__multiset,axiom,
! [M: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M @ X4 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_663_filter__preserves__multiset,axiom,
! [M: int > nat,P: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M @ X4 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_664_filter__preserves__multiset,axiom,
! [M: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M @ X4 ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M @ X4 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_665_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_666_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_667_ex__less__Suc,axiom,
! [J2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ J2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ J2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% ex_less_Suc
thf(fact_668_add__less__zeroD,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_669_add__less__zeroD,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_670_row__mat__of__row__fun,axiom,
! [I2: nat,Nr: nat,F: nat > vec_int,Nc: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ( dim_vec_int @ ( F @ I2 ) )
= Nc )
=> ( ( row_int @ ( mat_of_row_fun_int @ Nr @ Nc @ F ) @ I2 )
= ( F @ I2 ) ) ) ) ).
% row_mat_of_row_fun
thf(fact_671_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M6: nat] :
? [N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_672_dim__col__mat_I2_J,axiom,
! [Nr: nat,Nc: nat,G: nat > vec_int] :
( ( dim_col_int @ ( mat_of_row_fun_int @ Nr @ Nc @ G ) )
= Nc ) ).
% dim_col_mat(2)
thf(fact_673_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_674_finite__psubset__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ! [A5: set_int] :
( ( finite_finite_int @ A5 )
=> ( ! [B5: set_int] :
( ( ord_less_set_int @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_675_finite__psubset__induct,axiom,
! [A: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [A5: set_complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ! [B5: set_complex] :
( ( ord_less_set_complex @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_676_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_677_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_678_zero__one__matrix_Oin__map__col__valid__index,axiom,
! [Matrix: mat_int,I2: nat,C2: vec_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( member_nat @ I2 @ ( incide3973235006681262014ck_int @ C2 ) )
=> ( ord_less_nat @ I2 @ ( dim_vec_int @ C2 ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index
thf(fact_679_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N7: nat] :
( ( ord_less_nat @ M3 @ N7 )
& ( member_nat @ N7 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_680_map__col__to__block__alt,axiom,
( incide3975725477190312290ck_nat
= ( ^ [C4: vec_nat] :
( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ C4 ) )
& ( ( vec_index_nat @ C4 @ I )
= one_one_nat ) ) ) ) ) ).
% map_col_to_block_alt
thf(fact_681_map__col__to__block__alt,axiom,
( incide7996601054137363008omplex
= ( ^ [C4: vec_complex] :
( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_complex @ C4 ) )
& ( ( vec_index_complex @ C4 @ I )
= one_one_complex ) ) ) ) ) ).
% map_col_to_block_alt
thf(fact_682_map__col__to__block__alt,axiom,
( incide970706021007448894k_real
= ( ^ [C4: vec_real] :
( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_real @ C4 ) )
& ( ( vec_index_real @ C4 @ I )
= one_one_real ) ) ) ) ) ).
% map_col_to_block_alt
thf(fact_683_map__col__to__block__alt,axiom,
( incide3973235006681262014ck_int
= ( ^ [C4: vec_int] :
( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_int @ C4 ) )
& ( ( vec_index_int @ C4 @ I )
= one_one_int ) ) ) ) ) ).
% map_col_to_block_alt
thf(fact_684_map__col__to__block__elem,axiom,
! [I2: nat,C2: vec_nat] :
( ( ord_less_nat @ I2 @ ( dim_vec_nat @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide3975725477190312290ck_nat @ C2 ) )
= ( ( vec_index_nat @ C2 @ I2 )
= one_one_nat ) ) ) ).
% map_col_to_block_elem
thf(fact_685_map__col__to__block__elem,axiom,
! [I2: nat,C2: vec_complex] :
( ( ord_less_nat @ I2 @ ( dim_vec_complex @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide7996601054137363008omplex @ C2 ) )
= ( ( vec_index_complex @ C2 @ I2 )
= one_one_complex ) ) ) ).
% map_col_to_block_elem
thf(fact_686_map__col__to__block__elem,axiom,
! [I2: nat,C2: vec_real] :
( ( ord_less_nat @ I2 @ ( dim_vec_real @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide970706021007448894k_real @ C2 ) )
= ( ( vec_index_real @ C2 @ I2 )
= one_one_real ) ) ) ).
% map_col_to_block_elem
thf(fact_687_map__col__to__block__elem,axiom,
! [I2: nat,C2: vec_int] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide3973235006681262014ck_int @ C2 ) )
= ( ( vec_index_int @ C2 @ I2 )
= one_one_int ) ) ) ).
% map_col_to_block_elem
thf(fact_688_lift__01__vec__simp_I2_J,axiom,
! [I2: nat,V2: vec_int] :
( ( ord_less_nat @ I2 @ ( dim_vec_int @ V2 ) )
=> ( ( vec_index_int @ ( matrix8301520909418075407nt_int @ V2 ) @ I2 )
= ( matrix1697308990001484774nt_int @ ( vec_index_int @ V2 @ I2 ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_689_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
= ( P @ B2 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B2: nat] :
( ( P @ A2 @ B2 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_690_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_691_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_692_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_693_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_694_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_695_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_696_of__zero__neq__one__0__iff,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_697_of__zero__neq__one__0__iff,axiom,
! [X2: int] :
( ( ( matrix1699799460510535050nt_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_698_of__zero__neq__one__0__iff,axiom,
! [X2: complex] :
( ( ( matrix4491085277152940556ex_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_complex ) ) ).
% of_zero_neq_one_0_iff
thf(fact_699_of__zero__neq__one__0__iff,axiom,
! [X2: real] :
( ( ( matrix4086780077301154698al_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_700_of__zero__neq__one__0__iff,axiom,
! [X2: nat] :
( ( ( matrix697955278100430218at_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_701_of__zero__neq__one__0__iff,axiom,
! [X2: int] :
( ( ( matrix1697308990001484774nt_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_702_of__zero__neq__one__0__iff,axiom,
! [X2: complex] :
( ( ( matrix4488594806643890280ex_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_complex ) ) ).
% of_zero_neq_one_0_iff
thf(fact_703_of__zero__neq__one__0__iff,axiom,
! [X2: real] :
( ( ( matrix4084289606792104422al_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_704_of__zero__neq__one__0__iff,axiom,
! [X2: nat] :
( ( ( matrix871301952718202892omplex @ X2 )
= zero_zero_complex )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_705_of__zero__neq__one__0__iff,axiom,
! [X2: int] :
( ( ( matrix1846837417924380264omplex @ X2 )
= zero_zero_complex )
= ( X2 = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_706_of__zero__neq__one__0,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_707_of__zero__neq__one__0,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_708_of__zero__neq__one__0,axiom,
( ( matrix871301952718202892omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_zero_neq_one_0
thf(fact_709_of__zero__neq__one__0,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_710_of__zero__neq__one__0,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_711_of__zero__neq__one__0,axiom,
( ( matrix1697308990001484774nt_int @ zero_zero_int )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_712_of__zero__neq__one__0,axiom,
( ( matrix1846837417924380264omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_zero_neq_one_0
thf(fact_713_of__zero__neq__one__0,axiom,
( ( matrix1706393078865277798t_real @ zero_zero_int )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_714_of__zero__neq__one__0,axiom,
( ( matrix4491085277152940556ex_nat @ zero_zero_complex )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_715_of__zero__neq__one__0,axiom,
( ( matrix4488594806643890280ex_int @ zero_zero_complex )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_716_of__zero__hom_Ohom__0__iff,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_717_of__zero__hom_Ohom__0__iff,axiom,
! [X2: int] :
( ( ( matrix1699799460510535050nt_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_718_of__zero__hom_Ohom__0__iff,axiom,
! [X2: complex] :
( ( ( matrix4491085277152940556ex_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_complex ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_719_of__zero__hom_Ohom__0__iff,axiom,
! [X2: real] :
( ( ( matrix4086780077301154698al_nat @ X2 )
= zero_zero_nat )
= ( X2 = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_720_of__zero__hom_Ohom__0__iff,axiom,
! [X2: nat] :
( ( ( matrix697955278100430218at_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_721_of__zero__hom_Ohom__0__iff,axiom,
! [X2: int] :
( ( ( matrix1697308990001484774nt_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_722_of__zero__hom_Ohom__0__iff,axiom,
! [X2: complex] :
( ( ( matrix4488594806643890280ex_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_complex ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_723_of__zero__hom_Ohom__0__iff,axiom,
! [X2: real] :
( ( ( matrix4084289606792104422al_int @ X2 )
= zero_zero_int )
= ( X2 = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_724_of__zero__hom_Ohom__0__iff,axiom,
! [X2: nat] :
( ( ( matrix871301952718202892omplex @ X2 )
= zero_zero_complex )
= ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_725_of__zero__hom_Ohom__0__iff,axiom,
! [X2: int] :
( ( ( matrix1846837417924380264omplex @ X2 )
= zero_zero_complex )
= ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_726_of__zero__hom_Ohom__zero,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_727_of__zero__hom_Ohom__zero,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_728_of__zero__hom_Ohom__zero,axiom,
( ( matrix871301952718202892omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_zero_hom.hom_zero
thf(fact_729_of__zero__hom_Ohom__zero,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_730_of__zero__hom_Ohom__zero,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_731_of__zero__hom_Ohom__zero,axiom,
( ( matrix1697308990001484774nt_int @ zero_zero_int )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_732_of__zero__hom_Ohom__zero,axiom,
( ( matrix1846837417924380264omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_zero_hom.hom_zero
thf(fact_733_of__zero__hom_Ohom__zero,axiom,
( ( matrix1706393078865277798t_real @ zero_zero_int )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_734_of__zero__hom_Ohom__zero,axiom,
( ( matrix4491085277152940556ex_nat @ zero_zero_complex )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_735_of__zero__hom_Ohom__zero,axiom,
( ( matrix4488594806643890280ex_int @ zero_zero_complex )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_736_of__zero__neq__one__1,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_737_of__zero__neq__one__1,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_738_of__zero__neq__one__1,axiom,
( ( matrix871301952718202892omplex @ one_one_nat )
= one_one_complex ) ).
% of_zero_neq_one_1
thf(fact_739_of__zero__neq__one__1,axiom,
( ( matrix8742843541027031818t_real @ one_one_nat )
= one_one_real ) ).
% of_zero_neq_one_1
thf(fact_740_of__zero__neq__one__1,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_741_of__zero__neq__one__1,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_742_of__zero__neq__one__1,axiom,
( ( matrix1846837417924380264omplex @ one_one_int )
= one_one_complex ) ).
% of_zero_neq_one_1
thf(fact_743_of__zero__neq__one__1,axiom,
( ( matrix1706393078865277798t_real @ one_one_int )
= one_one_real ) ).
% of_zero_neq_one_1
thf(fact_744_of__zero__neq__one__1,axiom,
( ( matrix4491085277152940556ex_nat @ one_one_complex )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_745_of__zero__neq__one__1,axiom,
( ( matrix4488594806643890280ex_int @ one_one_complex )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_746_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_747_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_748_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix871301952718202892omplex @ one_one_nat )
= one_one_complex ) ).
% of_inj_on_01_hom.hom_one
thf(fact_749_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix8742843541027031818t_real @ one_one_nat )
= one_one_real ) ).
% of_inj_on_01_hom.hom_one
thf(fact_750_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_751_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_752_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1846837417924380264omplex @ one_one_int )
= one_one_complex ) ).
% of_inj_on_01_hom.hom_one
thf(fact_753_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1706393078865277798t_real @ one_one_int )
= one_one_real ) ).
% of_inj_on_01_hom.hom_one
thf(fact_754_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4491085277152940556ex_nat @ one_one_complex )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_755_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4488594806643890280ex_int @ one_one_complex )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_756_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_757_all__ones__index,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_nat @ ( matrix2751262895470517546ec_nat @ N ) @ I2 )
= one_one_nat ) ) ).
% all_ones_index
thf(fact_758_all__ones__index,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_int @ ( matrix2748772424961467270ec_int @ N ) @ I2 )
= one_one_int ) ) ).
% all_ones_index
thf(fact_759_all__ones__index,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_complex @ ( matrix5128154353246082568omplex @ N ) @ I2 )
= one_one_complex ) ) ).
% all_ones_index
thf(fact_760_all__ones__index,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_real @ ( matrix5166576126360777478c_real @ N ) @ I2 )
= one_one_real ) ) ).
% all_ones_index
thf(fact_761_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_762_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_763_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_764_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_765_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_766_one__reorient,axiom,
! [X2: complex] :
( ( one_one_complex = X2 )
= ( X2 = one_one_complex ) ) ).
% one_reorient
thf(fact_767_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_768_of__zero__neq__one__def,axiom,
( matrix700445748609480494at_nat
= ( ^ [X4: nat] : ( if_nat @ ( X4 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_769_of__zero__neq__one__def,axiom,
( matrix697955278100430218at_int
= ( ^ [X4: nat] : ( if_int @ ( X4 = zero_zero_nat ) @ zero_zero_int @ one_one_int ) ) ) ).
% of_zero_neq_one_def
thf(fact_770_of__zero__neq__one__def,axiom,
( matrix871301952718202892omplex
= ( ^ [X4: nat] : ( if_complex @ ( X4 = zero_zero_nat ) @ zero_zero_complex @ one_one_complex ) ) ) ).
% of_zero_neq_one_def
thf(fact_771_of__zero__neq__one__def,axiom,
( matrix8742843541027031818t_real
= ( ^ [X4: nat] : ( if_real @ ( X4 = zero_zero_nat ) @ zero_zero_real @ one_one_real ) ) ) ).
% of_zero_neq_one_def
thf(fact_772_of__zero__neq__one__def,axiom,
( matrix1699799460510535050nt_nat
= ( ^ [X4: int] : ( if_nat @ ( X4 = zero_zero_int ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_773_of__zero__neq__one__def,axiom,
( matrix1697308990001484774nt_int
= ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ one_one_int ) ) ) ).
% of_zero_neq_one_def
thf(fact_774_of__zero__neq__one__def,axiom,
( matrix1846837417924380264omplex
= ( ^ [X4: int] : ( if_complex @ ( X4 = zero_zero_int ) @ zero_zero_complex @ one_one_complex ) ) ) ).
% of_zero_neq_one_def
thf(fact_775_of__zero__neq__one__def,axiom,
( matrix1706393078865277798t_real
= ( ^ [X4: int] : ( if_real @ ( X4 = zero_zero_int ) @ zero_zero_real @ one_one_real ) ) ) ).
% of_zero_neq_one_def
thf(fact_776_of__zero__neq__one__def,axiom,
( matrix4491085277152940556ex_nat
= ( ^ [X4: complex] : ( if_nat @ ( X4 = zero_zero_complex ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% of_zero_neq_one_def
thf(fact_777_of__zero__neq__one__def,axiom,
( matrix4488594806643890280ex_int
= ( ^ [X4: complex] : ( if_int @ ( X4 = zero_zero_complex ) @ zero_zero_int @ one_one_int ) ) ) ).
% of_zero_neq_one_def
thf(fact_778_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_779_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_780_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_781_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_782_of__zero__hom_Ohom__0,axiom,
! [X2: nat] :
( ( ( matrix700445748609480494at_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_783_of__zero__hom_Ohom__0,axiom,
! [X2: int] :
( ( ( matrix1699799460510535050nt_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_784_of__zero__hom_Ohom__0,axiom,
! [X2: complex] :
( ( ( matrix4491085277152940556ex_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_complex ) ) ).
% of_zero_hom.hom_0
thf(fact_785_of__zero__hom_Ohom__0,axiom,
! [X2: real] :
( ( ( matrix4086780077301154698al_nat @ X2 )
= zero_zero_nat )
=> ( X2 = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_786_of__zero__hom_Ohom__0,axiom,
! [X2: nat] :
( ( ( matrix697955278100430218at_int @ X2 )
= zero_zero_int )
=> ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_787_of__zero__hom_Ohom__0,axiom,
! [X2: int] :
( ( ( matrix1697308990001484774nt_int @ X2 )
= zero_zero_int )
=> ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_788_of__zero__hom_Ohom__0,axiom,
! [X2: complex] :
( ( ( matrix4488594806643890280ex_int @ X2 )
= zero_zero_int )
=> ( X2 = zero_zero_complex ) ) ).
% of_zero_hom.hom_0
thf(fact_789_of__zero__hom_Ohom__0,axiom,
! [X2: real] :
( ( ( matrix4084289606792104422al_int @ X2 )
= zero_zero_int )
=> ( X2 = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_790_of__zero__hom_Ohom__0,axiom,
! [X2: nat] :
( ( ( matrix871301952718202892omplex @ X2 )
= zero_zero_complex )
=> ( X2 = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_791_of__zero__hom_Ohom__0,axiom,
! [X2: int] :
( ( ( matrix1846837417924380264omplex @ X2 )
= zero_zero_complex )
=> ( X2 = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_792_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_793_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_794_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_795_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_796_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_797_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_798_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_799_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_800_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_801_add__mono1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_802_add__mono1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_803_add__mono1,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% add_mono1
thf(fact_804_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_805_less__add__one,axiom,
! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).
% less_add_one
thf(fact_806_less__add__one,axiom,
! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).
% less_add_one
thf(fact_807_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_808_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_809_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_810_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_811_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_812_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_813_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_814_is__num__normalize_I1_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_815_is__num__normalize_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_816_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_817_sum__eq__1__iff,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A )
= one_one_nat )
= ( ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y4: complex] :
( ( member_complex @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_818_sum__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= one_one_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_819_sum__eq__1__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= one_one_nat )
= ( ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y4: int] :
( ( member_int @ Y4 @ A )
=> ( ( X4 != Y4 )
=> ( ( F @ Y4 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_820_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_int,I2: nat,C2: vec_nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I2 @ ( dim_vec_nat @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide3975725477190312290ck_nat @ C2 ) )
= ( ( vec_index_nat @ C2 @ I2 )
= one_one_nat ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_821_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_int,I2: nat,C2: vec_complex] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I2 @ ( dim_vec_complex @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide7996601054137363008omplex @ C2 ) )
= ( ( vec_index_complex @ C2 @ I2 )
= one_one_complex ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_822_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_int,I2: nat,C2: vec_real] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I2 @ ( dim_vec_real @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide970706021007448894k_real @ C2 ) )
= ( ( vec_index_real @ C2 @ I2 )
= one_one_real ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_823_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_int,I2: nat,C2: vec_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I2 @ ( dim_vec_int @ C2 ) )
=> ( ( member_nat @ I2 @ ( incide3973235006681262014ck_int @ C2 ) )
= ( ( vec_index_int @ C2 @ I2 )
= one_one_int ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_824_zero__one__matrix_Omap__col__to__block__alt,axiom,
! [Matrix: mat_int,C2: vec_nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide3975725477190312290ck_nat @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ C2 ) )
& ( ( vec_index_nat @ C2 @ I )
= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_alt
thf(fact_825_zero__one__matrix_Omap__col__to__block__alt,axiom,
! [Matrix: mat_int,C2: vec_complex] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide7996601054137363008omplex @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_complex @ C2 ) )
& ( ( vec_index_complex @ C2 @ I )
= one_one_complex ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_alt
thf(fact_826_zero__one__matrix_Omap__col__to__block__alt,axiom,
! [Matrix: mat_int,C2: vec_real] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide970706021007448894k_real @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_real @ C2 ) )
& ( ( vec_index_real @ C2 @ I )
= one_one_real ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_alt
thf(fact_827_zero__one__matrix_Omap__col__to__block__alt,axiom,
! [Matrix: mat_int,C2: vec_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide3973235006681262014ck_int @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) )
& ( ( vec_index_int @ C2 @ I )
= one_one_int ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_alt
thf(fact_828_row__nth__0__or__1__iff,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ m ) )
=> ( ( ( vec_index_int @ ( row_int @ m @ I2 ) @ J2 )
= zero_zero_int )
= ( ( vec_index_int @ ( row_int @ m @ I2 ) @ J2 )
!= one_one_int ) ) ) ) ).
% row_nth_0_or_1_iff
thf(fact_829_map__col__to__block__def,axiom,
( incide3975725477190312290ck_nat
= ( ^ [C4: vec_nat] :
( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_nat @ C4 ) ) )
& ( ( vec_index_nat @ C4 @ I )
= one_one_nat ) ) ) ) ) ).
% map_col_to_block_def
thf(fact_830_map__col__to__block__def,axiom,
( incide7996601054137363008omplex
= ( ^ [C4: vec_complex] :
( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_complex @ C4 ) ) )
& ( ( vec_index_complex @ C4 @ I )
= one_one_complex ) ) ) ) ) ).
% map_col_to_block_def
thf(fact_831_map__col__to__block__def,axiom,
( incide970706021007448894k_real
= ( ^ [C4: vec_real] :
( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_real @ C4 ) ) )
& ( ( vec_index_real @ C4 @ I )
= one_one_real ) ) ) ) ) ).
% map_col_to_block_def
thf(fact_832_map__col__to__block__def,axiom,
( incide3973235006681262014ck_int
= ( ^ [C4: vec_int] :
( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_int @ C4 ) ) )
& ( ( vec_index_int @ C4 @ I )
= one_one_int ) ) ) ) ) ).
% map_col_to_block_def
thf(fact_833_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_nat,J2: nat,I2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_nat @ Matrix ) )
=> ( ( ( vec_index_nat @ ( row_nat @ Matrix @ I2 ) @ J2 )
= zero_zero_nat )
= ( ( vec_index_nat @ ( row_nat @ Matrix @ I2 ) @ J2 )
!= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_834_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_complex,J2: nat,I2: nat] :
( ( incide5998224313882735548omplex @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_complex @ Matrix ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_complex @ Matrix ) )
=> ( ( ( vec_index_complex @ ( row_complex @ Matrix @ I2 ) @ J2 )
= zero_zero_complex )
= ( ( vec_index_complex @ ( row_complex @ Matrix @ I2 ) @ J2 )
!= one_one_complex ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_835_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_real,J2: nat,I2: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ Matrix ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_real @ Matrix ) )
=> ( ( ( vec_index_real @ ( row_real @ Matrix @ I2 ) @ J2 )
= zero_zero_real )
= ( ( vec_index_real @ ( row_real @ Matrix @ I2 ) @ J2 )
!= one_one_real ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_836_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_int,J2: nat,I2: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ Matrix ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ Matrix ) )
=> ( ( ( vec_index_int @ ( row_int @ Matrix @ I2 ) @ J2 )
= zero_zero_int )
= ( ( vec_index_int @ ( row_int @ Matrix @ I2 ) @ J2 )
!= one_one_int ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_837_row__add_I2_J,axiom,
! [I2: nat,A: mat_int,B: mat_int] :
( ( ord_less_nat @ I2 @ ( dim_row_int @ A ) )
=> ( ( ( dim_row_int @ B )
= ( dim_row_int @ A ) )
=> ( ( ( dim_col_int @ B )
= ( dim_col_int @ A ) )
=> ( ( row_int @ ( plus_plus_mat_int @ A @ B ) @ I2 )
= ( plus_plus_vec_int @ ( row_int @ A @ I2 ) @ ( row_int @ B @ I2 ) ) ) ) ) ) ).
% row_add(2)
thf(fact_838_index__unit__vec_I1_J,axiom,
! [I2: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I2 )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I2 ) @ J2 )
= one_one_nat ) )
& ( ( J2 != I2 )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I2 ) @ J2 )
= zero_zero_nat ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_839_index__unit__vec_I1_J,axiom,
! [I2: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I2 )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I2 ) @ J2 )
= one_one_int ) )
& ( ( J2 != I2 )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I2 ) @ J2 )
= zero_zero_int ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_840_index__unit__vec_I1_J,axiom,
! [I2: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I2 )
=> ( ( vec_index_complex @ ( unit_vec_complex @ N @ I2 ) @ J2 )
= one_one_complex ) )
& ( ( J2 != I2 )
=> ( ( vec_index_complex @ ( unit_vec_complex @ N @ I2 ) @ J2 )
= zero_zero_complex ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_841_index__unit__vec_I1_J,axiom,
! [I2: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I2 )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I2 ) @ J2 )
= one_one_real ) )
& ( ( J2 != I2 )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I2 ) @ J2 )
= zero_zero_real ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_842_lessThan__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_843_lessThan__iff,axiom,
! [I2: complex,K: complex] :
( ( member_complex @ I2 @ ( set_or7194820819169546315omplex @ K ) )
= ( ord_less_complex @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_844_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_845_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_846_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_847_index__add__mat_I2_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_row_int @ ( plus_plus_mat_int @ A @ B ) )
= ( dim_row_int @ B ) ) ).
% index_add_mat(2)
thf(fact_848_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_849_lift__01__mat__simp_I1_J,axiom,
! [M: mat_int] :
( ( dim_row_int @ ( matrix323868623736973467nt_int @ M ) )
= ( dim_row_int @ M ) ) ).
% lift_01_mat_simp(1)
thf(fact_850_all__ones__mat__dim__row,axiom,
! [N: nat] :
( ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_row
thf(fact_851_dim__row__mat_I2_J,axiom,
! [Nr: nat,Nc: nat,G: nat > vec_int] :
( ( dim_row_int @ ( mat_of_row_fun_int @ Nr @ Nc @ G ) )
= Nr ) ).
% dim_row_mat(2)
thf(fact_852_dim__update__mat_I1_J,axiom,
! [A: mat_int,Ij: product_prod_nat_nat,A3: int] :
( ( dim_row_int @ ( update_mat_int @ A @ Ij @ A3 ) )
= ( dim_row_int @ A ) ) ).
% dim_update_mat(1)
thf(fact_853_mat__of__row__dim_I1_J,axiom,
! [Y3: vec_int] :
( ( dim_row_int @ ( mat_of_row_int @ Y3 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_854_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_855_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_856_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_857_sum_OlessThan__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_858_index__unit__vec_I2_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I2 ) @ I2 )
= one_one_nat ) ) ).
% index_unit_vec(2)
thf(fact_859_index__unit__vec_I2_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I2 ) @ I2 )
= one_one_int ) ) ).
% index_unit_vec(2)
thf(fact_860_index__unit__vec_I2_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_complex @ ( unit_vec_complex @ N @ I2 ) @ I2 )
= one_one_complex ) ) ).
% index_unit_vec(2)
thf(fact_861_index__unit__vec_I2_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I2 ) @ I2 )
= one_one_real ) ) ).
% index_unit_vec(2)
thf(fact_862_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_863_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_864_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I5: int] :
( ( ord_less_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_865_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_866_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_867_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_868_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_869_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_870_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_871_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_872_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_873_infinite__Iio,axiom,
! [A3: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A3 ) ) ).
% infinite_Iio
thf(fact_874_lessThan__def,axiom,
( set_or7194820819169546315omplex
= ( ^ [U2: complex] :
( collect_complex
@ ^ [X4: complex] : ( ord_less_complex @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_875_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_876_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_real @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_877_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_878_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_879_lessThan__strict__subset__iff,axiom,
! [M2: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_880_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_881_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_882_eq__rowI,axiom,
! [B: mat_int,A: mat_int] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( dim_row_int @ B ) )
=> ( ( row_int @ A @ I5 )
= ( row_int @ B @ I5 ) ) )
=> ( ( ( dim_row_int @ A )
= ( dim_row_int @ B ) )
=> ( ( ( dim_col_int @ A )
= ( dim_col_int @ B ) )
=> ( A = B ) ) ) ) ).
% eq_rowI
thf(fact_883_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_884_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_885_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_886_sum_OlessThan__Suc__shift,axiom,
! [G: nat > complex,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( G @ zero_zero_nat )
@ ( groups2073611262835488442omplex
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_887_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_888_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_889_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_890_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > complex,N: nat] :
( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_891_zero__one__matrix_Omap__col__to__block__def,axiom,
! [Matrix: mat_int,C2: vec_nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide3975725477190312290ck_nat @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_nat @ C2 ) ) )
& ( ( vec_index_nat @ C2 @ I )
= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_def
thf(fact_892_zero__one__matrix_Omap__col__to__block__def,axiom,
! [Matrix: mat_int,C2: vec_complex] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide7996601054137363008omplex @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_complex @ C2 ) ) )
& ( ( vec_index_complex @ C2 @ I )
= one_one_complex ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_def
thf(fact_893_zero__one__matrix_Omap__col__to__block__def,axiom,
! [Matrix: mat_int,C2: vec_real] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide970706021007448894k_real @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_real @ C2 ) ) )
& ( ( vec_index_real @ C2 @ I )
= one_one_real ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_def
thf(fact_894_zero__one__matrix_Omap__col__to__block__def,axiom,
! [Matrix: mat_int,C2: vec_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( incide3973235006681262014ck_int @ C2 )
= ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ ( dim_vec_int @ C2 ) ) )
& ( ( vec_index_int @ C2 @ I )
= one_one_int ) ) ) ) ) ).
% zero_one_matrix.map_col_to_block_def
thf(fact_895_sum__bounds__lt__plus1,axiom,
! [F: nat > int,Mm: nat] :
( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_896_sum__bounds__lt__plus1,axiom,
! [F: nat > real,Mm: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_897_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_898_sum__bounds__lt__plus1,axiom,
! [F: nat > complex,Mm: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_899_col__nth__0__or__1__iff,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ m ) )
=> ( ( ( vec_index_int @ ( col_int @ m @ J2 ) @ I2 )
= zero_zero_int )
= ( ( vec_index_int @ ( col_int @ m @ J2 ) @ I2 )
!= one_one_int ) ) ) ) ).
% col_nth_0_or_1_iff
thf(fact_900_proper__inc__mat__def,axiom,
( incide294466202882093137at_int
= ( ^ [M7: mat_int] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_int @ M7 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_int @ M7 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_901_in__map__col__valid__index__M,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( member_nat @ I2 @ ( incide3973235006681262014ck_int @ ( col_int @ m @ J2 ) ) )
=> ( ord_less_nat @ I2 @ ( dim_row_int @ m ) ) ) ) ).
% in_map_col_valid_index_M
thf(fact_902_assms_I1_J,axiom,
ord_less_nat @ i @ ( dim_row_int @ ( times_times_mat_int @ m @ ( transpose_mat_int @ m ) ) ) ).
% assms(1)
thf(fact_903_Matrix_Otranspose__transpose,axiom,
! [A: mat_int] :
( ( transpose_mat_int @ ( transpose_mat_int @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_904_transpose__mat__eq,axiom,
! [A: mat_int,B: mat_int] :
( ( ( transpose_mat_int @ A )
= ( transpose_mat_int @ B ) )
= ( A = B ) ) ).
% transpose_mat_eq
thf(fact_905_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_906_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_907_mult__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_908_mult__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_909_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_910_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_911_mult__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B3 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_912_mult__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_913_mult__eq__0__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_914_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_915_mult__eq__0__iff,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_916_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_917_mult__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_918_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_919_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_920_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_921_mult__zero__left,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_922_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_923_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_924_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_925_mult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% mult_1
thf(fact_926_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_927_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_928_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_929_mult_Oright__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.right_neutral
thf(fact_930_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_931_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_932_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_933_index__mult__mat_I2_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_row_int @ ( times_times_mat_int @ A @ B ) )
= ( dim_row_int @ A ) ) ).
% index_mult_mat(2)
thf(fact_934_index__mult__mat_I3_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_col_int @ ( times_times_mat_int @ A @ B ) )
= ( dim_col_int @ B ) ) ).
% index_mult_mat(3)
thf(fact_935_finite__atLeastAtMost__int,axiom,
! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_936_finite__atLeastLessThan__int,axiom,
! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_937_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A3 @ I )
& ( ord_less_int @ I @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_938_mult__cancel__left1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_939_mult__cancel__left1,axiom,
! [C2: int,B3: int] :
( ( C2
= ( times_times_int @ C2 @ B3 ) )
= ( ( C2 = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_940_mult__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_941_mult__cancel__left2,axiom,
! [C2: complex,A3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_942_mult__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ( times_times_int @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_943_mult__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ( times_times_real @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_944_mult__cancel__right1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_945_mult__cancel__right1,axiom,
! [C2: int,B3: int] :
( ( C2
= ( times_times_int @ B3 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_946_mult__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_947_mult__cancel__right2,axiom,
! [A3: complex,C2: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_948_mult__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ( times_times_int @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_949_mult__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ( times_times_real @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_950_dim__col,axiom,
! [A: mat_int,I2: nat] :
( ( dim_vec_int @ ( col_int @ A @ I2 ) )
= ( dim_row_int @ A ) ) ).
% dim_col
thf(fact_951_index__transpose__mat_I2_J,axiom,
! [A: mat_int] :
( ( dim_row_int @ ( transpose_mat_int @ A ) )
= ( dim_col_int @ A ) ) ).
% index_transpose_mat(2)
thf(fact_952_index__transpose__mat_I3_J,axiom,
! [A: mat_int] :
( ( dim_col_int @ ( transpose_mat_int @ A ) )
= ( dim_row_int @ A ) ) ).
% index_transpose_mat(3)
thf(fact_953_col__transpose,axiom,
! [I2: nat,A: mat_int] :
( ( ord_less_nat @ I2 @ ( dim_row_int @ A ) )
=> ( ( col_int @ ( transpose_mat_int @ A ) @ I2 )
= ( row_int @ A @ I2 ) ) ) ).
% col_transpose
thf(fact_954_row__transpose,axiom,
! [J2: nat,A: mat_int] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ A ) )
=> ( ( row_int @ ( transpose_mat_int @ A ) @ J2 )
= ( col_int @ A @ J2 ) ) ) ).
% row_transpose
thf(fact_955_mult_Oleft__commute,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_956_mult_Oleft__commute,axiom,
! [B3: int,A3: int,C2: int] :
( ( times_times_int @ B3 @ ( times_times_int @ A3 @ C2 ) )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_957_mult_Oleft__commute,axiom,
! [B3: real,A3: real,C2: real] :
( ( times_times_real @ B3 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_958_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_959_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_960_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_961_mult_Oassoc,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_962_mult_Oassoc,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_963_mult_Oassoc,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_964_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_965_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_966_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_967_mult__not__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B3 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_968_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_969_mult__not__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B3 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_970_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_971_divisors__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_972_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_973_divisors__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_974_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_975_no__zero__divisors,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B3 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_976_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_977_no__zero__divisors,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_978_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_979_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_980_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_981_mult__left__cancel,axiom,
! [C2: int,A3: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_982_mult__left__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_983_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_984_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_985_mult__right__cancel,axiom,
! [C2: int,A3: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_986_mult__right__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_987_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X2: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_988_crossproduct__eq,axiom,
! [W: int,Y3: int,X2: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X2 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_989_crossproduct__eq,axiom,
! [W: real,Y3: real,X2: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X2 @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_990_crossproduct__noteq,axiom,
! [A3: nat,B3: nat,C2: nat,D: nat] :
( ( ( A3 != B3 )
& ( C2 != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A3 @ D ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_991_crossproduct__noteq,axiom,
! [A3: int,B3: int,C2: int,D: int] :
( ( ( A3 != B3 )
& ( C2 != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A3 @ D ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_992_crossproduct__noteq,axiom,
! [A3: real,B3: real,C2: real,D: real] :
( ( ( A3 != B3 )
& ( C2 != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A3 @ D ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_993_combine__common__factor,axiom,
! [A3: nat,E2: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_994_combine__common__factor,axiom,
! [A3: int,E2: int,B3: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_995_combine__common__factor,axiom,
! [A3: real,E2: real,B3: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_996_distrib__right,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_997_distrib__right,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_998_distrib__right,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_999_distrib__left,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_1000_distrib__left,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_1001_distrib__left,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_1002_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1003_comm__semiring__class_Odistrib,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1004_comm__semiring__class_Odistrib,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1005_ring__class_Oring__distribs_I1_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1006_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1007_ring__class_Oring__distribs_I2_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1008_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1009_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_1010_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_1011_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_1012_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_1013_comm__monoid__mult__class_Omult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1014_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1015_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1016_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1017_lambda__zero,axiom,
( ( ^ [H3: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_1018_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_1019_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_1020_lambda__zero,axiom,
( ( ^ [H3: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_1021_lambda__one,axiom,
( ( ^ [X4: complex] : X4 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_1022_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1023_lambda__one,axiom,
( ( ^ [X4: int] : X4 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1024_lambda__one,axiom,
( ( ^ [X4: real] : X4 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_1025_sum__product,axiom,
! [F: nat > int,A: set_nat,G: nat > int,B: set_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1026_sum__product,axiom,
! [F: nat > int,A: set_nat,G: int > int,B: set_int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups4538972089207619220nt_int @ G @ B ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups4538972089207619220nt_int
@ ^ [J: int] : ( times_times_int @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1027_sum__product,axiom,
! [F: complex > complex,A: set_complex,G: complex > complex,B: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ ( groups7754918857620584856omplex @ G @ B ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1028_sum__product,axiom,
! [F: complex > complex,A: set_complex,G: int > complex,B: set_int] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ ( groups3049146728041665814omplex @ G @ B ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups3049146728041665814omplex
@ ^ [J: int] : ( times_times_complex @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1029_sum__product,axiom,
! [F: complex > complex,A: set_complex,G: nat > complex,B: set_nat] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ ( groups2073611262835488442omplex @ G @ B ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups2073611262835488442omplex
@ ^ [J: nat] : ( times_times_complex @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1030_sum__product,axiom,
! [F: nat > real,A: set_nat,G: nat > real,B: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1031_sum__product,axiom,
! [F: nat > real,A: set_nat,G: int > real,B: set_int] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups8778361861064173332t_real @ G @ B ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups8778361861064173332t_real
@ ^ [J: int] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1032_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat
@ ^ [J: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1033_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: int > nat,B: set_int] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ G @ B ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups4541462559716669496nt_nat
@ ^ [J: int] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1034_sum__product,axiom,
! [F: int > real,A: set_int,G: nat > real,B: set_nat] :
( ( times_times_real @ ( groups8778361861064173332t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B ) )
= ( groups8778361861064173332t_real
@ ^ [I: int] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J ) )
@ B )
@ A ) ) ).
% sum_product
thf(fact_1035_double__sum__mult__hom,axiom,
! [K: int,F: nat > nat > int,G: nat > set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_int @ K
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups3539618377306564664at_int @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1036_double__sum__mult__hom,axiom,
! [K: int,F: nat > int > int,G: nat > set_int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] :
( groups4538972089207619220nt_int
@ ^ [J: int] : ( times_times_int @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_int @ K
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( groups4538972089207619220nt_int @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1037_double__sum__mult__hom,axiom,
! [K: complex,F: complex > complex > complex,G: complex > set_complex,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_complex @ K
@ ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1038_double__sum__mult__hom,axiom,
! [K: complex,F: complex > int > complex,G: complex > set_int,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups3049146728041665814omplex
@ ^ [J: int] : ( times_times_complex @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_complex @ K
@ ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups3049146728041665814omplex @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1039_double__sum__mult__hom,axiom,
! [K: complex,F: complex > nat > complex,G: complex > set_nat,A: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups2073611262835488442omplex
@ ^ [J: nat] : ( times_times_complex @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_complex @ K
@ ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups2073611262835488442omplex @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1040_double__sum__mult__hom,axiom,
! [K: real,F: nat > nat > real,G: nat > set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_real @ K
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1041_double__sum__mult__hom,axiom,
! [K: real,F: nat > int > real,G: nat > set_int,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups8778361861064173332t_real
@ ^ [J: int] : ( times_times_real @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_real @ K
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups8778361861064173332t_real @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1042_double__sum__mult__hom,axiom,
! [K: real,F: int > nat > real,G: int > set_nat,A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [I: int] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_real @ K
@ ( groups8778361861064173332t_real
@ ^ [I: int] : ( groups6591440286371151544t_real @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1043_double__sum__mult__hom,axiom,
! [K: real,F: int > int > real,G: int > set_int,A: set_int] :
( ( groups8778361861064173332t_real
@ ^ [I: int] :
( groups8778361861064173332t_real
@ ^ [J: int] : ( times_times_real @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_real @ K
@ ( groups8778361861064173332t_real
@ ^ [I: int] : ( groups8778361861064173332t_real @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1044_double__sum__mult__hom,axiom,
! [K: complex,F: int > complex > complex,G: int > set_complex,A: set_int] :
( ( groups3049146728041665814omplex
@ ^ [I: int] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ K @ ( F @ I @ J ) )
@ ( G @ I ) )
@ A )
= ( times_times_complex @ K
@ ( groups3049146728041665814omplex
@ ^ [I: int] : ( groups7754918857620584856omplex @ ( F @ I ) @ ( G @ I ) )
@ A ) ) ) ).
% double_sum_mult_hom
thf(fact_1045_sum__distrib__right,axiom,
! [F: nat > int,A: set_nat,R2: int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ R2 )
= ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1046_sum__distrib__right,axiom,
! [F: complex > complex,A: set_complex,R2: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1047_sum__distrib__right,axiom,
! [F: nat > real,A: set_nat,R2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1048_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R2: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ R2 )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1049_sum__distrib__right,axiom,
! [F: int > real,A: set_int,R2: real] :
( ( times_times_real @ ( groups8778361861064173332t_real @ F @ A ) @ R2 )
= ( groups8778361861064173332t_real
@ ^ [N3: int] : ( times_times_real @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1050_sum__distrib__right,axiom,
! [F: int > complex,A: set_int,R2: complex] :
( ( times_times_complex @ ( groups3049146728041665814omplex @ F @ A ) @ R2 )
= ( groups3049146728041665814omplex
@ ^ [N3: int] : ( times_times_complex @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1051_sum__distrib__right,axiom,
! [F: nat > complex,A: set_nat,R2: complex] :
( ( times_times_complex @ ( groups2073611262835488442omplex @ F @ A ) @ R2 )
= ( groups2073611262835488442omplex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1052_sum__distrib__right,axiom,
! [F: int > int,A: set_int,R2: int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A ) @ R2 )
= ( groups4538972089207619220nt_int
@ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1053_sum__distrib__right,axiom,
! [F: int > nat,A: set_int,R2: nat] :
( ( times_times_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ R2 )
= ( groups4541462559716669496nt_nat
@ ^ [N3: int] : ( times_times_nat @ ( F @ N3 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_1054_sum__distrib__left,axiom,
! [R2: int,F: nat > int,A: set_nat] :
( ( times_times_int @ R2 @ ( groups3539618377306564664at_int @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( times_times_int @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1055_sum__distrib__left,axiom,
! [R2: complex,F: complex > complex,A: set_complex] :
( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A ) )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1056_sum__distrib__left,axiom,
! [R2: real,F: nat > real,A: set_nat] :
( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1057_sum__distrib__left,axiom,
! [R2: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1058_sum__distrib__left,axiom,
! [R2: real,F: int > real,A: set_int] :
( ( times_times_real @ R2 @ ( groups8778361861064173332t_real @ F @ A ) )
= ( groups8778361861064173332t_real
@ ^ [N3: int] : ( times_times_real @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1059_sum__distrib__left,axiom,
! [R2: complex,F: int > complex,A: set_int] :
( ( times_times_complex @ R2 @ ( groups3049146728041665814omplex @ F @ A ) )
= ( groups3049146728041665814omplex
@ ^ [N3: int] : ( times_times_complex @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1060_sum__distrib__left,axiom,
! [R2: complex,F: nat > complex,A: set_nat] :
( ( times_times_complex @ R2 @ ( groups2073611262835488442omplex @ F @ A ) )
= ( groups2073611262835488442omplex
@ ^ [N3: nat] : ( times_times_complex @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1061_sum__distrib__left,axiom,
! [R2: int,F: int > int,A: set_int] :
( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A ) )
= ( groups4538972089207619220nt_int
@ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1062_sum__distrib__left,axiom,
! [R2: nat,F: int > nat,A: set_int] :
( ( times_times_nat @ R2 @ ( groups4541462559716669496nt_nat @ F @ A ) )
= ( groups4541462559716669496nt_nat
@ ^ [N3: int] : ( times_times_nat @ R2 @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_1063_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1064_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1065_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1066_mult__less__cancel__right__disj,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1067_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1068_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1069_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1070_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1071_mult__strict__right__mono__neg,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1072_mult__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1073_mult__less__cancel__left__disj,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1074_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1075_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1076_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1077_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1078_mult__strict__left__mono__neg,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1079_mult__strict__left__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1080_mult__less__cancel__left__pos,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1081_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1082_mult__less__cancel__left__neg,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1083_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1084_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1085_zero__less__mult__pos2,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1086_zero__less__mult__pos2,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1087_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_1088_zero__less__mult__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_1089_zero__less__mult__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_1090_zero__less__mult__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1091_zero__less__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1092_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1093_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1094_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1095_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1096_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1097_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1098_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1099_block__nempty__implies__all__zeros,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( ( incide3973235006681262014ck_int @ ( col_int @ m @ J2 ) )
= bot_bot_set_nat )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ m ) )
=> ( ( vec_index_int @ ( col_int @ m @ J2 ) @ I2 )
= zero_zero_int ) ) ) ) ).
% block_nempty_implies_all_zeros
thf(fact_1100_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_1101_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1109_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1110_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1111_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1112_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1113_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1114_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times_int @ zero_zero_int @ L2 )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1115_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1116_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1117_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1118_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1119_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1120_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1121_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1122_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1123_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1124_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1125_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_1126_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_1127_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1128_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1129_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1130_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1131_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1132_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1133_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_1134_one__implies__block__nempty,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( member_int @ one_one_int @ ( vec_set_int @ ( col_int @ m @ J2 ) ) )
=> ( ( incide3973235006681262014ck_int @ ( col_int @ m @ J2 ) )
!= bot_bot_set_nat ) ) ) ).
% one_implies_block_nempty
thf(fact_1135_block__nempty__implies__no__one,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( ( incide3973235006681262014ck_int @ ( col_int @ m @ J2 ) )
= bot_bot_set_nat )
=> ~ ( member_int @ one_one_int @ ( vec_set_int @ ( col_int @ m @ J2 ) ) ) ) ) ).
% block_nempty_implies_no_one
thf(fact_1136_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_1137_non__empty__col__01,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_int @ m ) )
=> ( ( incide6851923868969248411ol_int @ m @ J2 )
= ( member_int @ one_one_int @ ( vec_set_int @ ( col_int @ m @ J2 ) ) ) ) ) ).
% non_empty_col_01
thf(fact_1138_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1139_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_1140_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1141_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_1142_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_1143_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_1144_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M2: nat] :
( ( ( power_power_nat @ X2 @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1145_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1146_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1147_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1148_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_1149_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1150_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C2 ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1151_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
& ( ( power_power_real @ X @ N )
= A3 )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N )
= A3 ) )
=> ( Y6 = X ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1152_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1153_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1154_real__arch__pow,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N2: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1155_real__arch__pow__inv,axiom,
! [Y3: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N2 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1156_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_1157_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_1158_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_1159_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1160_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1161_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_1162_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_1163_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_1164_less__mult__imp__div__less,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1165_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1166_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1167_div__neg__pos__less0,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1168_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_1169_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_1170_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_1171_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1172_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I: nat,J: nat] :
( ( ( ord_less_nat @ J @ N )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J ) ) )
=> ( P @ I ) ) ) ) ) ).
% split_div
thf(fact_1173_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1174_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1175_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_1176_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_1177_Maclaurin__lemma,axiom,
! [H: real,F: real > real,J2: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H )
=> ? [B6: real] :
( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J2 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B6 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_1178_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_1179_ln__less__cancel__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
= ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1180_ln__inj__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% ln_inj_iff
thf(fact_1181_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1182_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1183_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1184_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= zero_zero_real )
= ( X2 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1185_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1186_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1187_ln__less__self,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_1188_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_1189_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1190_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1191_ln__mult,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
= ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% ln_mult
thf(fact_1192_polynomial__product__nat,axiom,
! [M2: nat,A3: nat > nat,N: nat,B3: nat > nat,X2: nat] :
( ! [I5: nat] :
( ( ord_less_nat @ M2 @ I5 )
=> ( ( A3 @ I5 )
= zero_zero_nat ) )
=> ( ! [J3: nat] :
( ( ord_less_nat @ N @ J3 )
=> ( ( B3 @ J3 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( times_times_nat @ ( A3 @ I ) @ ( power_power_nat @ X2 @ I ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( times_times_nat @ ( B3 @ J ) @ ( power_power_nat @ X2 @ J ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R4: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( A3 @ K2 ) @ ( B3 @ ( minus_minus_nat @ R4 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R4 ) )
@ ( power_power_nat @ X2 @ R4 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_1193_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1194_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1195_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1196_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1197_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1198_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1199_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1200_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1201_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1202_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1203_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1204_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1205_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1206_diff__less__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1207_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1208_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1209_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1210_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1211_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1212_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1213_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1214_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1215_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1216_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1217_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1218_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1219_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1220_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1221_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1222_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1223_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ( ( ord_less_nat @ A3 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1224_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1225_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1226_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1227_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1228_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1229_fact__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_1230_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M2 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1231_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_1232_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1233_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1234_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1235_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1236_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1237_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1238_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1239_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1240_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1241_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1242_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_1243_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_1244_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1245_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1246_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1247_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1248_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1249_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1250_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1251_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1252_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1253_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1254_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1255_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1256_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1257_diff__less__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1258_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1259_int__ops_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1260_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1261_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y3: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
= ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
& ( ( ord_less_nat @ X2 @ Y3 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1262_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I5: int] :
( ( ord_less_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1263_eq__diff__eq_H,axiom,
! [X2: real,Y3: real,Z2: real] :
( ( X2
= ( minus_minus_real @ Y3 @ Z2 ) )
= ( Y3
= ( plus_plus_real @ X2 @ Z2 ) ) ) ).
% eq_diff_eq'
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $true @ X2 @ Y3 )
= 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,Y3: complex] :
( ( if_complex @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y3: complex] :
( ( if_complex @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( matrix3634415343793898042ec_int @ ( row_int @ m @ i ) )
= ( groups3539618377306564664at_int @ ( vec_index_int @ ( row_int @ m @ i ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_col_int @ m ) ) ) ) ).
%------------------------------------------------------------------------------