TPTP Problem File: SLH0622^1.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 : LP_Duality/0001_LP_Duality/prob_00025_000776__28721166_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1435 ( 442 unt; 167 typ; 0 def)
% Number of atoms : 4312 (1241 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12630 ( 347 ~; 231 |; 251 &;9760 @)
% ( 0 <=>;2041 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 712 ( 712 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 141 usr; 16 con; 0-3 aty)
% Number of variables : 3985 ( 385 ^;3560 !; 40 ?;3985 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:55:33.685
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Real__Oreal_J_J,type,
set_vec_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
set_vec_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
set_vec_int: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
vec_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $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__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $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 ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (141)
thf(sy_c_Conjugate_Oconjugate__class_Oconjugate_001t__Matrix__Ovec_It__Int__Oint_J,type,
conjug4884081946748880444ec_int: vec_int > vec_int ).
thf(sy_c_Conjugate_Oconjugate__class_Oconjugate_001t__Matrix__Ovec_It__Real__Oreal_J,type,
conjug3612589096773959740c_real: vec_real > vec_real ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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__Int__Oint,type,
plus_plus_int: int > int > int ).
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__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__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
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_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
times_1230744552615602198_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_Itf__a_J,type,
times_times_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
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__Polynomial__Opoly_It__Int__Oint_J,type,
zero_zero_poly_int: poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Omat_Itf__a_J_001t__Int__Oint,type,
groups6532580691654848579_a_int: ( mat_a > int ) > set_mat_a > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Omat_Itf__a_J_001t__Nat__Onat,type,
groups6535071162163898855_a_nat: ( mat_a > nat ) > set_mat_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Omat_Itf__a_J_001t__Real__Oreal,type,
groups5751486073070743875a_real: ( mat_a > real ) > set_mat_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Ovec_Itf__a_J_001t__Int__Oint,type,
groups3782466442477828303_a_int: ( vec_a > int ) > set_vec_a > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Ovec_Itf__a_J_001t__Nat__Onat,type,
groups3784956912986878579_a_nat: ( vec_a > nat ) > set_vec_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Matrix__Ovec_Itf__a_J_001t__Real__Oreal,type,
groups7515830029211251663a_real: ( vec_a > real ) > set_vec_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001tf__a,type,
groups1143116142660632562_nat_a: ( nat > a ) > set_nat > a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001tf__a,type,
groups3370478925210225046real_a: ( real > a ) > set_real > a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Int__Oint,type,
carrier_vec_int: nat > set_vec_int ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Nat__Onat,type,
carrier_vec_nat: nat > set_vec_nat ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Real__Oreal,type,
carrier_vec_real: nat > set_vec_real ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
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_Odim__vec_001t__Set__Oset_It__Nat__Onat_J,type,
dim_vec_set_nat: vec_set_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001tf__a,type,
dim_vec_a: vec_a > nat ).
thf(sy_c_Matrix_Omat__diag_001t__Int__Oint,type,
mat_diag_int: nat > ( nat > int ) > mat_int ).
thf(sy_c_Matrix_Omat__diag_001t__Nat__Onat,type,
mat_diag_nat: nat > ( nat > nat ) > mat_nat ).
thf(sy_c_Matrix_Omat__diag_001t__Real__Oreal,type,
mat_diag_real: nat > ( nat > real ) > mat_real ).
thf(sy_c_Matrix_Omat__diag_001tf__a,type,
mat_diag_a: nat > ( nat > a ) > mat_a ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Real__Oreal,type,
mult_mat_vec_real: mat_real > vec_real > vec_real ).
thf(sy_c_Matrix_Omult__mat__vec_001tf__a,type,
mult_mat_vec_a: mat_a > vec_a > vec_a ).
thf(sy_c_Matrix_Oscalar__prod_001t__Int__Oint,type,
scalar_prod_int: vec_int > vec_int > int ).
thf(sy_c_Matrix_Oscalar__prod_001t__Nat__Onat,type,
scalar_prod_nat: vec_nat > vec_nat > nat ).
thf(sy_c_Matrix_Oscalar__prod_001t__Real__Oreal,type,
scalar_prod_real: vec_real > vec_real > real ).
thf(sy_c_Matrix_Oscalar__prod_001tf__a,type,
scalar_prod_a: vec_a > vec_a > a ).
thf(sy_c_Matrix_Otranspose__mat_001t__Real__Oreal,type,
transpose_mat_real: mat_real > mat_real ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__vec_001tf__a,type,
update_vec_a: vec_a > nat > a > vec_a ).
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__index_001t__Set__Oset_It__Nat__Onat_J,type,
vec_index_set_nat: vec_set_nat > nat > set_nat ).
thf(sy_c_Matrix_Ovec__index_001tf__a,type,
vec_index_a: vec_a > nat > a ).
thf(sy_c_Matrix_Ozero__vec_001t__Int__Oint,type,
zero_vec_int: nat > vec_int ).
thf(sy_c_Matrix_Ozero__vec_001t__Nat__Onat,type,
zero_vec_nat: nat > vec_nat ).
thf(sy_c_Matrix_Ozero__vec_001t__Real__Oreal,type,
zero_vec_real: nat > vec_real ).
thf(sy_c_Matrix_Ozero__vec_001tf__a,type,
zero_vec_a: nat > vec_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Norms_Olinf__norm__vec_001t__Int__Oint,type,
linf_norm_vec_int: vec_int > int ).
thf(sy_c_Norms_Olinf__norm__vec_001t__Real__Oreal,type,
linf_norm_vec_real: vec_real > real ).
thf(sy_c_Norms_Onorm1_001t__Int__Oint,type,
norm1_int: poly_int > int ).
thf(sy_c_Norms_Onorm1_001t__Real__Oreal,type,
norm1_real: poly_real > real ).
thf(sy_c_Norms_Osq__norm__poly_001t__Int__Oint,type,
sq_norm_poly_int: poly_int > int ).
thf(sy_c_Norms_Osq__norm__poly_001t__Real__Oreal,type,
sq_norm_poly_real: poly_real > real ).
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__Matrix__Omat_Itf__a_J,type,
ord_less_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Ovec_Itf__a_J,type,
ord_less_vec_a: vec_a > vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_Itf__a_J_M_Eo_J,type,
ord_less_eq_mat_a_o: ( mat_a > $o ) > ( mat_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Ovec_Itf__a_J_M_Eo_J,type,
ord_less_eq_vec_a_o: ( vec_a > $o ) > ( vec_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_Itf__a_J,type,
ord_less_eq_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Int__Oint_J,type,
ord_less_eq_vec_int: vec_int > vec_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_eq_vec_nat: vec_nat > vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Real__Oreal_J,type,
ord_less_eq_vec_real: vec_real > vec_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le2754941374173645459et_nat: vec_set_nat > vec_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_Itf__a_J,type,
ord_less_eq_vec_a: vec_a > vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__a_J,type,
collect_vec_a: ( vec_a > $o ) > set_vec_a ).
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_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Matrix__Omat_Itf__a_J,type,
set_or1377778852321182218_mat_a: mat_a > mat_a > set_mat_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Matrix__Ovec_Itf__a_J,type,
set_or2357829874413910678_vec_a: vec_a > vec_a > set_vec_a ).
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_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Int__Oint_J,type,
member_vec_int: vec_int > set_vec_int > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat: vec_nat > set_vec_nat > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Real__Oreal_J,type,
member_vec_real: vec_real > set_vec_real > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a: vec_a > set_vec_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_b,type,
b: vec_a ).
thf(sy_v_c,type,
c: vec_a ).
thf(sy_v_nc,type,
nc: nat ).
thf(sy_v_nr,type,
nr: nat ).
thf(sy_v_x,type,
x: vec_a ).
thf(sy_v_y,type,
y: vec_a ).
% Relevant facts (1267)
thf(fact_0_c,axiom,
member_vec_a @ c @ ( carrier_vec_a @ nc ) ).
% c
thf(fact_1_A,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ nr @ nc ) ).
% A
thf(fact_2_y0,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ y ).
% y0
thf(fact_3_b,axiom,
member_vec_a @ b @ ( carrier_vec_a @ nr ) ).
% b
thf(fact_4_x,axiom,
member_vec_a @ x @ ( carrier_vec_a @ nc ) ).
% x
thf(fact_5_Axb,axiom,
ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ x ) @ b ).
% Axb
thf(fact_6_y,axiom,
member_vec_a @ y @ ( carrier_vec_a @ nr ) ).
% y
thf(fact_7_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_mat_a] :
( ( groups6535071162163898855_a_nat
@ ^ [I: mat_a] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% class_semiring.add.finprod_one
thf(fact_8_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_mat_a] :
( ( groups6532580691654848579_a_int
@ ^ [I: mat_a] : zero_zero_int
@ A )
= zero_zero_int ) ).
% class_semiring.add.finprod_one
thf(fact_9_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% class_semiring.add.finprod_one
thf(fact_10_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_real] :
( ( groups1935376822645274424al_nat
@ ^ [I: real] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% class_semiring.add.finprod_one
thf(fact_11_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [I: real] : zero_zero_int
@ A )
= zero_zero_int ) ).
% class_semiring.add.finprod_one
thf(fact_12_class__semiring_Oadd_Ofinprod__one,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% class_semiring.add.finprod_one
thf(fact_13_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_real,F: real > nat] :
( ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_nat ) )
=> ( ( groups1935376822645274424al_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_all1
thf(fact_14_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_real,F: real > int] :
( ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_int ) )
=> ( ( groups1932886352136224148al_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_all1
thf(fact_15_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_real,F: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_all1
thf(fact_16_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_nat,F: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_all1
thf(fact_17_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_vec_a,F: vec_a > nat] :
( ! [A2: vec_a] :
( ( member_vec_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_nat ) )
=> ( ( groups3784956912986878579_a_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_all1
thf(fact_18_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_mat_a,F: mat_a > nat] :
( ! [A2: mat_a] :
( ( member_mat_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_nat ) )
=> ( ( groups6535071162163898855_a_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_all1
thf(fact_19_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_vec_a,F: vec_a > int] :
( ! [A2: vec_a] :
( ( member_vec_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_int ) )
=> ( ( groups3782466442477828303_a_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_all1
thf(fact_20_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_mat_a,F: mat_a > int] :
( ! [A2: mat_a] :
( ( member_mat_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_int ) )
=> ( ( groups6532580691654848579_a_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_all1
thf(fact_21_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_vec_a,F: vec_a > real] :
( ! [A2: vec_a] :
( ( member_vec_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_real ) )
=> ( ( groups7515830029211251663a_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_all1
thf(fact_22_class__semiring_Oadd_Ofinprod__all1,axiom,
! [A: set_mat_a,F: mat_a > real] :
( ! [A2: mat_a] :
( ( member_mat_a @ A2 @ A )
=> ( ( F @ A2 )
= zero_zero_real ) )
=> ( ( groups5751486073070743875a_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_all1
thf(fact_23_sum_Oneutral__const,axiom,
! [A: set_mat_a] :
( ( groups6535071162163898855_a_nat
@ ^ [Uu: mat_a] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_24_sum_Oneutral__const,axiom,
! [A: set_mat_a] :
( ( groups6532580691654848579_a_int
@ ^ [Uu: mat_a] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_25_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [Uu: real] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_26_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups1935376822645274424al_nat
@ ^ [Uu: real] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_27_sum_Oneutral__const,axiom,
! [A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [Uu: real] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_28_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups1143116142660632562_nat_a
@ ^ [Uu: nat] : zero_zero_a
@ A )
= zero_zero_a ) ).
% sum.neutral_const
thf(fact_29_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_30_ivl__subset,axiom,
! [I2: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I2 @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I2 )
| ( ( ord_less_eq_real @ M @ I2 )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_31_ivl__subset,axiom,
! [I2: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I2 @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I2 )
| ( ( ord_less_eq_int @ M @ I2 )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_32_ivl__subset,axiom,
! [I2: a,J: a,M: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M @ N ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M @ I2 )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_33_ivl__subset,axiom,
! [I2: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_34_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_35_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_36_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_37_mult__zero__left,axiom,
! [A3: a] :
( ( times_times_a @ zero_zero_a @ A3 )
= zero_zero_a ) ).
% mult_zero_left
thf(fact_38_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_39_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_40_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_41_mult__zero__right,axiom,
! [A3: a] :
( ( times_times_a @ A3 @ zero_zero_a )
= zero_zero_a ) ).
% mult_zero_right
thf(fact_42_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_43_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_44_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_45_mult__eq__0__iff,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_46_mult__eq__0__iff,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_47_mult__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_48_mult__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_49_mult__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_50_mult__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_51_mult__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_52_mult__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_53_mult__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_54_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_55_yA,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ y )
= c ) ).
% yA
thf(fact_56_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_57_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_58_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_59_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_60_mult_Oleft__commute,axiom,
! [B: a,A3: a,C: a] :
( ( times_times_a @ B @ ( times_times_a @ A3 @ C ) )
= ( times_times_a @ A3 @ ( times_times_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_61_mult_Oleft__commute,axiom,
! [B: nat,A3: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A3 @ C ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_62_mult_Oleft__commute,axiom,
! [B: int,A3: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A3 @ C ) )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_63_mult_Oleft__commute,axiom,
! [B: real,A3: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A3 @ C ) )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_64_mult_Ocommute,axiom,
( times_times_a
= ( ^ [A4: a,B2: a] : ( times_times_a @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_65_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B2: nat] : ( times_times_nat @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_66_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B2: int] : ( times_times_int @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_67_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B2: real] : ( times_times_real @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_68_mult_Oassoc,axiom,
! [A3: a,B: a,C: a] :
( ( times_times_a @ ( times_times_a @ A3 @ B ) @ C )
= ( times_times_a @ A3 @ ( times_times_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_69_mult_Oassoc,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_70_mult_Oassoc,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_71_mult_Oassoc,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_72_class__cring_Ofactors__equal,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( A3 = B )
=> ( ( C = D )
=> ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_73_class__cring_Ofactors__equal,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( A3 = B )
=> ( ( C = D )
=> ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_74_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: a,B: a,C: a] :
( ( times_times_a @ ( times_times_a @ A3 @ B ) @ C )
= ( times_times_a @ A3 @ ( times_times_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_75_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_76_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B ) @ C )
= ( times_times_int @ A3 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_77_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B ) @ C )
= ( times_times_real @ A3 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_78_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J: real > nat,T: set_nat,H: nat > a,G: real > a] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_nat @ ( J @ A2 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3370478925210225046real_a @ G @ S )
= ( groups1143116142660632562_nat_a @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_79_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_real @ ( J @ A2 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_80_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J: nat > real,T: set_real,H: real > int,G: nat > int] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_real @ ( J @ A2 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups1932886352136224148al_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_81_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J: nat > real,T: set_real,H: real > a,G: nat > a] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_real @ ( J @ A2 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1143116142660632562_nat_a @ G @ S )
= ( groups3370478925210225046real_a @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_82_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,H: nat > a,G: nat > a] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_nat @ ( J @ A2 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1143116142660632562_nat_a @ G @ S )
= ( groups1143116142660632562_nat_a @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_83_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_nat @ ( J @ A2 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_84_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J: nat > real,T: set_real,H: real > nat,G: nat > nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( member_real @ ( J @ A2 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_85_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_nat @ ( J @ A2 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_86_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: real > real,J: real > real,T: set_real,H: real > real,G: real > real] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_real @ ( J @ A2 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_87_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J: real > nat,T: set_nat,H: nat > nat,G: real > nat] :
( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( I2 @ ( J @ A2 ) )
= A2 ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( member_nat @ ( J @ A2 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A2: real] :
( ( member_real @ A2 @ S )
=> ( ( H @ ( J @ A2 ) )
= ( G @ A2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_88_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > a,Phi: real > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_real @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3370478925210225046real_a @ Phi @ A )
= ( groups1143116142660632562_nat_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_89_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_90_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > int,Phi: nat > int] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_91_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > a,Phi: nat > a] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1143116142660632562_nat_a @ Phi @ A )
= ( groups3370478925210225046real_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_92_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1143116142660632562_nat_a @ Phi @ A )
= ( groups1143116142660632562_nat_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_93_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_94_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups1935376822645274424al_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_95_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_real @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_96_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > real,A: set_real,H: real > real,Gamma: real > real,Phi: real > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_real @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_97_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > real,A: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_real @ ( K @ Y ) @ A )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_98_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_real,H: real > nat,Gamma: nat > a,Phi: real > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3370478925210225046real_a @ Phi @ A )
= ( groups1143116142660632562_nat_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_99_sum_Oeq__general,axiom,
! [B4: set_real,A: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_100_sum_Oeq__general,axiom,
! [B4: set_real,A: set_nat,H: nat > real,Gamma: real > int,Phi: nat > int] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups1932886352136224148al_int @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_101_sum_Oeq__general,axiom,
! [B4: set_real,A: set_nat,H: nat > real,Gamma: real > a,Phi: nat > a] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1143116142660632562_nat_a @ Phi @ A )
= ( groups3370478925210225046real_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_102_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_nat,H: nat > nat,Gamma: nat > a,Phi: nat > a] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1143116142660632562_nat_a @ Phi @ A )
= ( groups1143116142660632562_nat_a @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_103_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_104_sum_Oeq__general,axiom,
! [B4: set_real,A: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups1935376822645274424al_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_105_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_106_sum_Oeq__general,axiom,
! [B4: set_real,A: set_real,H: real > real,Gamma: real > real,Phi: real > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_107_sum_Oeq__general,axiom,
! [B4: set_nat,A: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ( H @ X3 )
= Y )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B4 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_108_sum_Ocong,axiom,
! [A: set_nat,B4: set_nat,G: nat > a,H: nat > a] :
( ( A = B4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1143116142660632562_nat_a @ G @ A )
= ( groups1143116142660632562_nat_a @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_109_sum_Ocong,axiom,
! [A: set_nat,B4: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_110_sum_Ocong,axiom,
! [A: set_mat_a,B4: set_mat_a,G: mat_a > nat,H: mat_a > nat] :
( ( A = B4 )
=> ( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6535071162163898855_a_nat @ G @ A )
= ( groups6535071162163898855_a_nat @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_111_sum_Ocong,axiom,
! [A: set_mat_a,B4: set_mat_a,G: mat_a > int,H: mat_a > int] :
( ( A = B4 )
=> ( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6532580691654848579_a_int @ G @ A )
= ( groups6532580691654848579_a_int @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_112_sum_Ocong,axiom,
! [A: set_real,B4: set_real,G: real > real,H: real > real] :
( ( A = B4 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ A )
= ( groups8097168146408367636l_real @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_113_sum_Ocong,axiom,
! [A: set_real,B4: set_real,G: real > nat,H: real > nat] :
( ( A = B4 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ A )
= ( groups1935376822645274424al_nat @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_114_sum_Ocong,axiom,
! [A: set_real,B4: set_real,G: real > int,H: real > int] :
( ( A = B4 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B4 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ A )
= ( groups1932886352136224148al_int @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_115_sum_Oswap,axiom,
! [G: nat > nat > a,B4: set_nat,A: set_nat] :
( ( groups1143116142660632562_nat_a
@ ^ [I: nat] : ( groups1143116142660632562_nat_a @ ( G @ I ) @ B4 )
@ A )
= ( groups1143116142660632562_nat_a
@ ^ [J2: nat] :
( groups1143116142660632562_nat_a
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_116_sum_Oswap,axiom,
! [G: nat > nat > nat,B4: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_117_sum_Oswap,axiom,
! [G: nat > real > nat,B4: set_real,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups1935376822645274424al_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups1935376822645274424al_nat
@ ^ [J2: real] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_118_sum_Oswap,axiom,
! [G: real > real > real,B4: set_real,A: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : ( groups8097168146408367636l_real @ ( G @ I ) @ B4 )
@ A )
= ( groups8097168146408367636l_real
@ ^ [J2: real] :
( groups8097168146408367636l_real
@ ^ [I: real] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_119_sum_Oswap,axiom,
! [G: real > nat > nat,B4: set_nat,A: set_real] :
( ( groups1935376822645274424al_nat
@ ^ [I: real] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups1935376822645274424al_nat
@ ^ [I: real] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_120_sum_Oswap,axiom,
! [G: real > real > nat,B4: set_real,A: set_real] :
( ( groups1935376822645274424al_nat
@ ^ [I: real] : ( groups1935376822645274424al_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups1935376822645274424al_nat
@ ^ [J2: real] :
( groups1935376822645274424al_nat
@ ^ [I: real] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_121_sum_Oswap,axiom,
! [G: real > real > int,B4: set_real,A: set_real] :
( ( groups1932886352136224148al_int
@ ^ [I: real] : ( groups1932886352136224148al_int @ ( G @ I ) @ B4 )
@ A )
= ( groups1932886352136224148al_int
@ ^ [J2: real] :
( groups1932886352136224148al_int
@ ^ [I: real] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_122_sum_Oswap,axiom,
! [G: nat > mat_a > nat,B4: set_mat_a,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups6535071162163898855_a_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups6535071162163898855_a_nat
@ ^ [J2: mat_a] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_123_sum_Oswap,axiom,
! [G: mat_a > nat > nat,B4: set_nat,A: set_mat_a] :
( ( groups6535071162163898855_a_nat
@ ^ [I: mat_a] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups6535071162163898855_a_nat
@ ^ [I: mat_a] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_124_sum_Oswap,axiom,
! [G: mat_a > real > nat,B4: set_real,A: set_mat_a] :
( ( groups6535071162163898855_a_nat
@ ^ [I: mat_a] : ( groups1935376822645274424al_nat @ ( G @ I ) @ B4 )
@ A )
= ( groups1935376822645274424al_nat
@ ^ [J2: real] :
( groups6535071162163898855_a_nat
@ ^ [I: mat_a] : ( G @ I @ J2 )
@ A )
@ B4 ) ) ).
% sum.swap
thf(fact_125_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_126_mult__right__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_127_mult__right__cancel,axiom,
! [C: int,A3: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C )
= ( times_times_int @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_128_mult__right__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_129_mult__left__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_130_mult__left__cancel,axiom,
! [C: int,A3: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A3 )
= ( times_times_int @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_131_mult__left__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_132_no__zero__divisors,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_133_no__zero__divisors,axiom,
! [A3: int,B: int] :
( ( A3 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A3 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_134_no__zero__divisors,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A3 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_135_divisors__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_136_divisors__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_137_divisors__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_138_mult__not__zero,axiom,
! [A3: a,B: a] :
( ( ( times_times_a @ A3 @ B )
!= zero_zero_a )
=> ( ( A3 != zero_zero_a )
& ( B != zero_zero_a ) ) ) ).
% mult_not_zero
thf(fact_139_mult__not__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_140_mult__not__zero,axiom,
! [A3: int,B: int] :
( ( ( times_times_int @ A3 @ B )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_141_mult__not__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_142_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_143_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_144_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > a,A: set_real] :
( ( ( groups3370478925210225046real_a @ G @ A )
!= zero_zero_a )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_145_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > a,A: set_nat] :
( ( ( groups1143116142660632562_nat_a @ G @ A )
!= zero_zero_a )
=> ~ ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_146_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_147_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_148_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_149_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: real] :
( ( member_real @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_150_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: vec_a > nat,A: set_vec_a] :
( ( ( groups3784956912986878579_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A2: vec_a] :
( ( member_vec_a @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_151_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: vec_a > int,A: set_vec_a] :
( ( ( groups3782466442477828303_a_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A2: vec_a] :
( ( member_vec_a @ A2 @ A )
=> ( ( G @ A2 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_152_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_153_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_154_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_155_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_156_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) )
=> ( ( groups1935376822645274424al_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_157_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) )
=> ( ( groups1932886352136224148al_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_158_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_vec_a,F: vec_a > nat] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) )
=> ( ( groups3784956912986878579_a_nat @ F @ A )
= zero_zero_nat ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_159_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_vec_a,F: vec_a > int] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) )
=> ( ( groups3782466442477828303_a_int @ F @ A )
= zero_zero_int ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_160_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_vec_a,F: vec_a > real] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_real ) )
=> ( ( groups7515830029211251663a_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_161_class__semiring_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_mat_a,F: mat_a > real] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_real ) )
=> ( ( groups5751486073070743875a_real @ F @ A )
= zero_zero_real ) ) ).
% class_semiring.add.finprod_one_eqI
thf(fact_162_sum_Oneutral,axiom,
! [A: set_nat,G: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_a ) )
=> ( ( groups1143116142660632562_nat_a @ G @ A )
= zero_zero_a ) ) ).
% sum.neutral
thf(fact_163_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_164_sum_Oneutral,axiom,
! [A: set_mat_a,G: mat_a > nat] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups6535071162163898855_a_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_165_sum_Oneutral,axiom,
! [A: set_mat_a,G: mat_a > int] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups6532580691654848579_a_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_166_sum_Oneutral,axiom,
! [A: set_real,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_167_sum_Oneutral,axiom,
! [A: set_real,G: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups1935376822645274424al_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_168_sum_Oneutral,axiom,
! [A: set_real,G: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups1932886352136224148al_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_169_atLeastLessThan__subset__iff,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A3 @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A3 )
| ( ( ord_less_eq_real @ C @ A3 )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_170_atLeastLessThan__subset__iff,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A3 @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A3 )
| ( ( ord_less_eq_int @ C @ A3 )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_171_atLeastLessThan__subset__iff,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A3 @ B ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B @ A3 )
| ( ( ord_less_eq_a @ C @ A3 )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_172_atLeastLessThan__subset__iff,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A3 @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A3 )
| ( ( ord_less_eq_nat @ C @ A3 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_173_lambda__zero,axiom,
( ( ^ [H2: a] : zero_zero_a )
= ( times_times_a @ zero_zero_a ) ) ).
% lambda_zero
thf(fact_174_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_175_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_176_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_177_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_178_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_179_sum__mono,axiom,
! [K2: set_real,F: real > a,G: real > a] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_a @ ( groups3370478925210225046real_a @ F @ K2 ) @ ( groups3370478925210225046real_a @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_180_sum__mono,axiom,
! [K2: set_nat,F: nat > a,G: nat > a] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_a @ ( groups1143116142660632562_nat_a @ F @ K2 ) @ ( groups1143116142660632562_nat_a @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_181_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_182_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_183_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_184_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_185_sum__mono,axiom,
! [K2: set_vec_a,F: vec_a > real,G: vec_a > real] :
( ! [I3: vec_a] :
( ( member_vec_a @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups7515830029211251663a_real @ F @ K2 ) @ ( groups7515830029211251663a_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_186_sum__mono,axiom,
! [K2: set_mat_a,F: mat_a > real,G: mat_a > real] :
( ! [I3: mat_a] :
( ( member_mat_a @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups5751486073070743875a_real @ F @ K2 ) @ ( groups5751486073070743875a_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_187_sum__product,axiom,
! [F: nat > a,A: set_nat,G: nat > a,B4: set_nat] :
( ( times_times_a @ ( groups1143116142660632562_nat_a @ F @ A ) @ ( groups1143116142660632562_nat_a @ G @ B4 ) )
= ( groups1143116142660632562_nat_a
@ ^ [I: nat] :
( groups1143116142660632562_nat_a
@ ^ [J2: nat] : ( times_times_a @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_188_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B4: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_189_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: real > nat,B4: set_real] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ G @ B4 ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups1935376822645274424al_nat
@ ^ [J2: real] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_190_sum__product,axiom,
! [F: real > real,A: set_real,G: real > real,B4: set_real] :
( ( times_times_real @ ( groups8097168146408367636l_real @ F @ A ) @ ( groups8097168146408367636l_real @ G @ B4 ) )
= ( groups8097168146408367636l_real
@ ^ [I: real] :
( groups8097168146408367636l_real
@ ^ [J2: real] : ( times_times_real @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_191_sum__product,axiom,
! [F: real > nat,A: set_real,G: nat > nat,B4: set_nat] :
( ( times_times_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
= ( groups1935376822645274424al_nat
@ ^ [I: real] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_192_sum__product,axiom,
! [F: real > nat,A: set_real,G: real > nat,B4: set_real] :
( ( times_times_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ G @ B4 ) )
= ( groups1935376822645274424al_nat
@ ^ [I: real] :
( groups1935376822645274424al_nat
@ ^ [J2: real] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_193_sum__product,axiom,
! [F: real > int,A: set_real,G: real > int,B4: set_real] :
( ( times_times_int @ ( groups1932886352136224148al_int @ F @ A ) @ ( groups1932886352136224148al_int @ G @ B4 ) )
= ( groups1932886352136224148al_int
@ ^ [I: real] :
( groups1932886352136224148al_int
@ ^ [J2: real] : ( times_times_int @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_194_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: mat_a > nat,B4: set_mat_a] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups6535071162163898855_a_nat @ G @ B4 ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups6535071162163898855_a_nat
@ ^ [J2: mat_a] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_195_sum__product,axiom,
! [F: mat_a > nat,A: set_mat_a,G: nat > nat,B4: set_nat] :
( ( times_times_nat @ ( groups6535071162163898855_a_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
= ( groups6535071162163898855_a_nat
@ ^ [I: mat_a] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_196_sum__product,axiom,
! [F: mat_a > nat,A: set_mat_a,G: real > nat,B4: set_real] :
( ( times_times_nat @ ( groups6535071162163898855_a_nat @ F @ A ) @ ( groups1935376822645274424al_nat @ G @ B4 ) )
= ( groups6535071162163898855_a_nat
@ ^ [I: mat_a] :
( groups1935376822645274424al_nat
@ ^ [J2: real] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B4 )
@ A ) ) ).
% sum_product
thf(fact_197_sum__distrib__right,axiom,
! [F: nat > a,A: set_nat,R: a] :
( ( times_times_a @ ( groups1143116142660632562_nat_a @ F @ A ) @ R )
= ( groups1143116142660632562_nat_a
@ ^ [N2: nat] : ( times_times_a @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_198_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_199_sum__distrib__right,axiom,
! [F: mat_a > nat,A: set_mat_a,R: nat] :
( ( times_times_nat @ ( groups6535071162163898855_a_nat @ F @ A ) @ R )
= ( groups6535071162163898855_a_nat
@ ^ [N2: mat_a] : ( times_times_nat @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_200_sum__distrib__right,axiom,
! [F: mat_a > int,A: set_mat_a,R: int] :
( ( times_times_int @ ( groups6532580691654848579_a_int @ F @ A ) @ R )
= ( groups6532580691654848579_a_int
@ ^ [N2: mat_a] : ( times_times_int @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_201_sum__distrib__right,axiom,
! [F: real > real,A: set_real,R: real] :
( ( times_times_real @ ( groups8097168146408367636l_real @ F @ A ) @ R )
= ( groups8097168146408367636l_real
@ ^ [N2: real] : ( times_times_real @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_202_sum__distrib__right,axiom,
! [F: real > nat,A: set_real,R: nat] :
( ( times_times_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ R )
= ( groups1935376822645274424al_nat
@ ^ [N2: real] : ( times_times_nat @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_203_sum__distrib__right,axiom,
! [F: real > int,A: set_real,R: int] :
( ( times_times_int @ ( groups1932886352136224148al_int @ F @ A ) @ R )
= ( groups1932886352136224148al_int
@ ^ [N2: real] : ( times_times_int @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_204_mem__Collect__eq,axiom,
! [A3: vec_a,P: vec_a > $o] :
( ( member_vec_a @ A3 @ ( collect_vec_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_205_mem__Collect__eq,axiom,
! [A3: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A3 @ ( collect_mat_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_206_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_207_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_208_Collect__mem__eq,axiom,
! [A: set_vec_a] :
( ( collect_vec_a
@ ^ [X4: vec_a] : ( member_vec_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_209_Collect__mem__eq,axiom,
! [A: set_mat_a] :
( ( collect_mat_a
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_210_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_211_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_212_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_213_sum__distrib__left,axiom,
! [R: a,F: nat > a,A: set_nat] :
( ( times_times_a @ R @ ( groups1143116142660632562_nat_a @ F @ A ) )
= ( groups1143116142660632562_nat_a
@ ^ [N2: nat] : ( times_times_a @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_214_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_215_sum__distrib__left,axiom,
! [R: nat,F: mat_a > nat,A: set_mat_a] :
( ( times_times_nat @ R @ ( groups6535071162163898855_a_nat @ F @ A ) )
= ( groups6535071162163898855_a_nat
@ ^ [N2: mat_a] : ( times_times_nat @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_216_sum__distrib__left,axiom,
! [R: int,F: mat_a > int,A: set_mat_a] :
( ( times_times_int @ R @ ( groups6532580691654848579_a_int @ F @ A ) )
= ( groups6532580691654848579_a_int
@ ^ [N2: mat_a] : ( times_times_int @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_217_sum__distrib__left,axiom,
! [R: real,F: real > real,A: set_real] :
( ( times_times_real @ R @ ( groups8097168146408367636l_real @ F @ A ) )
= ( groups8097168146408367636l_real
@ ^ [N2: real] : ( times_times_real @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_218_sum__distrib__left,axiom,
! [R: nat,F: real > nat,A: set_real] :
( ( times_times_nat @ R @ ( groups1935376822645274424al_nat @ F @ A ) )
= ( groups1935376822645274424al_nat
@ ^ [N2: real] : ( times_times_nat @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_219_sum__distrib__left,axiom,
! [R: int,F: real > int,A: set_real] :
( ( times_times_int @ R @ ( groups1932886352136224148al_int @ F @ A ) )
= ( groups1932886352136224148al_int
@ ^ [N2: real] : ( times_times_int @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_220_mult__hom_Ohom__sum,axiom,
! [C: nat,F: nat > nat,X5: set_nat] :
( ( times_times_nat @ C @ ( groups3542108847815614940at_nat @ F @ X5 ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( times_times_nat @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_221_mult__hom_Ohom__sum,axiom,
! [C: nat,F: mat_a > nat,X5: set_mat_a] :
( ( times_times_nat @ C @ ( groups6535071162163898855_a_nat @ F @ X5 ) )
= ( groups6535071162163898855_a_nat
@ ^ [X4: mat_a] : ( times_times_nat @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_222_mult__hom_Ohom__sum,axiom,
! [C: int,F: mat_a > int,X5: set_mat_a] :
( ( times_times_int @ C @ ( groups6532580691654848579_a_int @ F @ X5 ) )
= ( groups6532580691654848579_a_int
@ ^ [X4: mat_a] : ( times_times_int @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_223_mult__hom_Ohom__sum,axiom,
! [C: real,F: real > real,X5: set_real] :
( ( times_times_real @ C @ ( groups8097168146408367636l_real @ F @ X5 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( times_times_real @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_224_mult__hom_Ohom__sum,axiom,
! [C: nat,F: real > nat,X5: set_real] :
( ( times_times_nat @ C @ ( groups1935376822645274424al_nat @ F @ X5 ) )
= ( groups1935376822645274424al_nat
@ ^ [X4: real] : ( times_times_nat @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_225_mult__hom_Ohom__sum,axiom,
! [C: int,F: real > int,X5: set_real] :
( ( times_times_int @ C @ ( groups1932886352136224148al_int @ F @ X5 ) )
= ( groups1932886352136224148al_int
@ ^ [X4: real] : ( times_times_int @ C @ ( F @ X4 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_226_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_227_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_228_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_229_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_230_zero__le__mult__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_231_zero__le__mult__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_232_mult__nonneg__nonpos2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_233_mult__nonneg__nonpos2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_234_mult__nonneg__nonpos2,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_235_mult__nonneg__nonpos2,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ B @ A3 ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_236_mult__nonpos__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_237_mult__nonpos__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_238_mult__nonpos__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_239_mult__nonpos__nonneg,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ A3 @ zero_zero_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ B ) @ zero_zero_a ) ) ) ).
% mult_nonpos_nonneg
thf(fact_240_mult__nonneg__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_241_mult__nonneg__nonpos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_242_mult__nonneg__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_243_mult__nonneg__nonpos,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ B ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos
thf(fact_244_mult__nonneg__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_245_mult__nonneg__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_246_mult__nonneg__nonneg,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_247_mult__nonneg__nonneg,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_248_split__mult__neg__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_249_split__mult__neg__le,axiom,
! [A3: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_250_split__mult__neg__le,axiom,
! [A3: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_251_split__mult__neg__le,axiom,
! [A3: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A3 )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A3 @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ B ) @ zero_zero_a ) ) ).
% split_mult_neg_le
thf(fact_252_mult__le__0__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_253_mult__le__0__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_254_mult__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_255_mult__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_256_mult__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_257_mult__right__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_258_mult__right__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_259_mult__right__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_260_mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_261_mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_262_mult__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_263_mult__left__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_264_mult__nonpos__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_265_mult__nonpos__nonpos,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_266_mult__left__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_267_mult__left__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_268_split__mult__pos__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_269_split__mult__pos__le,axiom,
! [A3: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_270_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_271_zero__le__square,axiom,
! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_272_mult__mono_H,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_273_mult__mono_H,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_274_mult__mono_H,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_275_mult__mono_H,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_276_mult__mono,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_277_mult__mono,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_278_mult__mono,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_279_mult__mono,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_280_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_281_sum__nonpos,axiom,
! [A: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_282_sum__nonpos,axiom,
! [A: set_real,F: real > a] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_a @ ( F @ X2 ) @ zero_zero_a ) )
=> ( ord_less_eq_a @ ( groups3370478925210225046real_a @ F @ A ) @ zero_zero_a ) ) ).
% sum_nonpos
thf(fact_283_sum__nonpos,axiom,
! [A: set_nat,F: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_a @ ( F @ X2 ) @ zero_zero_a ) )
=> ( ord_less_eq_a @ ( groups1143116142660632562_nat_a @ F @ A ) @ zero_zero_a ) ) ).
% sum_nonpos
thf(fact_284_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_285_sum__nonpos,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_286_sum__nonpos,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_287_sum__nonpos,axiom,
! [A: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_288_sum__nonpos,axiom,
! [A: set_vec_a,F: vec_a > real] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups7515830029211251663a_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_289_sum__nonpos,axiom,
! [A: set_mat_a,F: mat_a > real] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5751486073070743875a_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_290_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_291_sum__nonneg,axiom,
! [A: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_292_sum__nonneg,axiom,
! [A: set_real,F: real > a] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_a @ zero_zero_a @ ( F @ X2 ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( groups3370478925210225046real_a @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_293_sum__nonneg,axiom,
! [A: set_nat,F: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_a @ zero_zero_a @ ( F @ X2 ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( groups1143116142660632562_nat_a @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_294_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_295_sum__nonneg,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_296_sum__nonneg,axiom,
! [A: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_297_sum__nonneg,axiom,
! [A: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_298_sum__nonneg,axiom,
! [A: set_vec_a,F: vec_a > real] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups7515830029211251663a_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_299_sum__nonneg,axiom,
! [A: set_mat_a,F: mat_a > real] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5751486073070743875a_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_300__092_060open_062c_A_092_060bullet_062_Ax_A_061_A_IA_092_060_094sup_062T_A_K_092_060_094sub_062v_Ay_J_A_092_060bullet_062_Ax_092_060close_062,axiom,
( ( scalar_prod_a @ c @ x )
= ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ y ) @ x ) ) ).
% \<open>c \<bullet> x = (A\<^sup>T *\<^sub>v y) \<bullet> x\<close>
thf(fact_301_assoc__mult__mat__vec,axiom,
! [A: mat_a,N_1: nat,N_2: nat,B4: mat_a,N_3: nat,V: vec_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N_3 ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ A @ B4 ) @ V )
= ( mult_mat_vec_a @ A @ ( mult_mat_vec_a @ B4 @ V ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_302_vec__of__dim__0,axiom,
! [V: vec_a] :
( ( ( dim_vec_a @ V )
= zero_zero_nat )
= ( V
= ( zero_vec_a @ zero_zero_nat ) ) ) ).
% vec_of_dim_0
thf(fact_303_calculation,axiom,
( ( scalar_prod_a @ c @ x )
= ( scalar_prod_a @ y @ ( mult_mat_vec_a @ a2 @ x ) ) ) ).
% calculation
thf(fact_304__092_060open_062_IA_092_060_094sup_062T_A_K_092_060_094sub_062v_Ay_J_A_092_060bullet_062_Ax_A_061_Ay_A_092_060bullet_062_A_IA_A_K_092_060_094sub_062v_Ax_J_092_060close_062,axiom,
( ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ y ) @ x )
= ( scalar_prod_a @ y @ ( mult_mat_vec_a @ a2 @ x ) ) ) ).
% \<open>(A\<^sup>T *\<^sub>v y) \<bullet> x = y \<bullet> (A *\<^sub>v x)\<close>
thf(fact_305_transpose__carrier__mat,axiom,
! [A: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_306_index__zero__vec_I2_J,axiom,
! [N: nat] :
( ( dim_vec_a @ ( zero_vec_a @ N ) )
= N ) ).
% index_zero_vec(2)
thf(fact_307_carrier__vecD,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( dim_vec_a @ V )
= N ) ) ).
% carrier_vecD
thf(fact_308_mult__mat__vec__carrier,axiom,
! [A: mat_a,Nr: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A @ V ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_309_square__lesseq__square,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ) ).
% square_lesseq_square
thf(fact_310_less__eq__fract__respect,axiom,
! [B: real,B5: real,D: real,D2: real,A3: real,A5: real,C: real,C2: real] :
( ( B != zero_zero_real )
=> ( ( B5 != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( D2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B5 )
= ( times_times_real @ A5 @ B ) )
=> ( ( ( times_times_real @ C @ D2 )
= ( times_times_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D ) @ ( times_times_real @ B @ D ) ) @ ( times_times_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ B @ D ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A5 @ D2 ) @ ( times_times_real @ B5 @ D2 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B5 ) @ ( times_times_real @ B5 @ D2 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_311_less__eq__fract__respect,axiom,
! [B: int,B5: int,D: int,D2: int,A3: int,A5: int,C: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( B5 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ B5 )
= ( times_times_int @ A5 @ B ) )
=> ( ( ( times_times_int @ C @ D2 )
= ( times_times_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A3 @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B5 ) @ ( times_times_int @ B5 @ D2 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_312_transpose__mat__eq,axiom,
! [A: mat_a,B4: mat_a] :
( ( ( transpose_mat_a @ A )
= ( transpose_mat_a @ B4 ) )
= ( A = B4 ) ) ).
% transpose_mat_eq
thf(fact_313_Matrix_Otranspose__transpose,axiom,
! [A: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_314_assoc__mult__mat,axiom,
! [A: mat_a,N_1: nat,N_2: nat,B4: mat_a,N_3: nat,C3: mat_a,N_4: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_mat_a @ C3 @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A @ B4 ) @ C3 )
= ( times_times_mat_a @ A @ ( times_times_mat_a @ B4 @ C3 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_315_scalar__prod__right__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V @ ( zero_vec_nat @ N ) )
= zero_zero_nat ) ) ).
% scalar_prod_right_zero
thf(fact_316_scalar__prod__right__zero,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ V @ ( zero_vec_int @ N ) )
= zero_zero_int ) ) ).
% scalar_prod_right_zero
thf(fact_317_scalar__prod__right__zero,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ V @ ( zero_vec_real @ N ) )
= zero_zero_real ) ) ).
% scalar_prod_right_zero
thf(fact_318_scalar__prod__right__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V @ ( zero_vec_a @ N ) )
= zero_zero_a ) ) ).
% scalar_prod_right_zero
thf(fact_319_scalar__prod__left__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( zero_vec_nat @ N ) @ V )
= zero_zero_nat ) ) ).
% scalar_prod_left_zero
thf(fact_320_scalar__prod__left__zero,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ ( zero_vec_int @ N ) @ V )
= zero_zero_int ) ) ).
% scalar_prod_left_zero
thf(fact_321_scalar__prod__left__zero,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ ( zero_vec_real @ N ) @ V )
= zero_zero_real ) ) ).
% scalar_prod_left_zero
thf(fact_322_scalar__prod__left__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( zero_vec_a @ N ) @ V )
= zero_zero_a ) ) ).
% scalar_prod_left_zero
thf(fact_323_comm__scalar__prod,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ V_2 )
= ( scalar_prod_a @ V_2 @ V_1 ) ) ) ) ).
% comm_scalar_prod
thf(fact_324_transpose__vec__mult__scalar,axiom,
! [A: mat_a,Nr: nat,Nc: nat,X: vec_a,Y2: vec_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ Y2 @ ( carrier_vec_a @ Nr ) )
=> ( ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ A ) @ Y2 ) @ X )
= ( scalar_prod_a @ Y2 @ ( mult_mat_vec_a @ A @ X ) ) ) ) ) ) ).
% transpose_vec_mult_scalar
thf(fact_325_mult__carrier__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B4: mat_a,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B4 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_326_scalar__prod__def,axiom,
( scalar_prod_int
= ( ^ [V2: vec_int,W: vec_int] :
( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( vec_index_int @ V2 @ I ) @ ( vec_index_int @ W @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_int @ W ) ) ) ) ) ).
% scalar_prod_def
thf(fact_327_scalar__prod__def,axiom,
( scalar_prod_real
= ( ^ [V2: vec_real,W: vec_real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( vec_index_real @ V2 @ I ) @ ( vec_index_real @ W @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_real @ W ) ) ) ) ) ).
% scalar_prod_def
thf(fact_328_scalar__prod__def,axiom,
( scalar_prod_a
= ( ^ [V2: vec_a,W: vec_a] :
( groups1143116142660632562_nat_a
@ ^ [I: nat] : ( times_times_a @ ( vec_index_a @ V2 @ I ) @ ( vec_index_a @ W @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_a @ W ) ) ) ) ) ).
% scalar_prod_def
thf(fact_329_scalar__prod__def,axiom,
( scalar_prod_nat
= ( ^ [V2: vec_nat,W: vec_nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( times_times_nat @ ( vec_index_nat @ V2 @ I ) @ ( vec_index_nat @ W @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_nat @ W ) ) ) ) ) ).
% scalar_prod_def
thf(fact_330_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_a @ ( zero_vec_a @ N ) @ ( carrier_vec_a @ N ) ) ).
% zero_carrier_vec
thf(fact_331_carrier__vecI,axiom,
! [V: vec_a,N: nat] :
( ( ( dim_vec_a @ V )
= N )
=> ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ).
% carrier_vecI
thf(fact_332_carrier__dim__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
= ( ( dim_vec_a @ V )
= N ) ) ).
% carrier_dim_vec
thf(fact_333_carrier__vec__dim__vec,axiom,
! [V: vec_a] : ( member_vec_a @ V @ ( carrier_vec_a @ ( dim_vec_a @ V ) ) ) ).
% carrier_vec_dim_vec
thf(fact_334_transpose__mult,axiom,
! [A: mat_a,Nr: nat,N: nat,B4: mat_a,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( transpose_mat_a @ ( times_times_mat_a @ A @ B4 ) )
= ( times_times_mat_a @ ( transpose_mat_a @ B4 ) @ ( transpose_mat_a @ A ) ) ) ) ) ).
% transpose_mult
thf(fact_335_carrier__vec__def,axiom,
( carrier_vec_a
= ( ^ [N2: nat] :
( collect_vec_a
@ ^ [V2: vec_a] :
( ( dim_vec_a @ V2 )
= N2 ) ) ) ) ).
% carrier_vec_def
thf(fact_336_gram__schmidt_OFarkas__Lemma,axiom,
! [A: mat_real,N: nat,Nr: nat,B: vec_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ Nr ) )
=> ( ( member_vec_real @ B @ ( carrier_vec_real @ N ) )
=> ( ( ? [X4: vec_real] :
( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ X4 )
& ( ( mult_mat_vec_real @ A @ X4 )
= B ) ) )
= ( ! [Y3: vec_real] :
( ( member_vec_real @ Y3 @ ( carrier_vec_real @ N ) )
=> ( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ ( mult_mat_vec_real @ ( transpose_mat_real @ A ) @ Y3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ Y3 @ B ) ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma
thf(fact_337_gram__schmidt_OFarkas__Lemma_H,axiom,
! [A: mat_real,Nr: nat,Nc: nat,B: vec_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_vec_real @ B @ ( carrier_vec_real @ Nr ) )
=> ( ( ? [X4: vec_real] :
( ( member_vec_real @ X4 @ ( carrier_vec_real @ Nc ) )
& ( ord_less_eq_vec_real @ ( mult_mat_vec_real @ A @ X4 ) @ B ) ) )
= ( ! [Y3: vec_real] :
( ( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ Y3 )
& ( ( mult_mat_vec_real @ ( transpose_mat_real @ A ) @ Y3 )
= ( zero_vec_real @ Nc ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ Y3 @ B ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma'
thf(fact_338_scalar__prod__ge__0,axiom,
! [X: vec_real] : ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_339_scalar__prod__ge__0,axiom,
! [X: vec_int] : ( ord_less_eq_int @ zero_zero_int @ ( scalar_prod_int @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_340_set__times__mono2,axiom,
! [C3: set_nat,D3: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ D3 )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( times_times_set_nat @ C3 @ E ) @ ( times_times_set_nat @ D3 @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_341_subset__antisym,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_342_subsetI,axiom,
! [A: set_vec_a,B4: set_vec_a] :
( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ A )
=> ( member_vec_a @ X2 @ B4 ) )
=> ( ord_le4791951621262958845_vec_a @ A @ B4 ) ) ).
% subsetI
thf(fact_343_subsetI,axiom,
! [A: set_mat_a,B4: set_mat_a] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( member_mat_a @ X2 @ B4 ) )
=> ( ord_le3318621148231462513_mat_a @ A @ B4 ) ) ).
% subsetI
thf(fact_344_subsetI,axiom,
! [A: set_real,B4: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B4 ) )
=> ( ord_less_eq_set_real @ A @ B4 ) ) ).
% subsetI
thf(fact_345_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_346_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_347_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_348_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_349_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_350_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_351_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_352_set__times__intro,axiom,
! [A3: mat_a,C3: set_mat_a,B: mat_a,D3: set_mat_a] :
( ( member_mat_a @ A3 @ C3 )
=> ( ( member_mat_a @ B @ D3 )
=> ( member_mat_a @ ( times_times_mat_a @ A3 @ B ) @ ( times_1230744552615602198_mat_a @ C3 @ D3 ) ) ) ) ).
% set_times_intro
thf(fact_353_set__times__intro,axiom,
! [A3: a,C3: set_a,B: a,D3: set_a] :
( ( member_a @ A3 @ C3 )
=> ( ( member_a @ B @ D3 )
=> ( member_a @ ( times_times_a @ A3 @ B ) @ ( times_times_set_a @ C3 @ D3 ) ) ) ) ).
% set_times_intro
thf(fact_354_set__times__intro,axiom,
! [A3: nat,C3: set_nat,B: nat,D3: set_nat] :
( ( member_nat @ A3 @ C3 )
=> ( ( member_nat @ B @ D3 )
=> ( member_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_set_nat @ C3 @ D3 ) ) ) ) ).
% set_times_intro
thf(fact_355_set__times__intro,axiom,
! [A3: int,C3: set_int,B: int,D3: set_int] :
( ( member_int @ A3 @ C3 )
=> ( ( member_int @ B @ D3 )
=> ( member_int @ ( times_times_int @ A3 @ B ) @ ( times_times_set_int @ C3 @ D3 ) ) ) ) ).
% set_times_intro
thf(fact_356_set__times__intro,axiom,
! [A3: real,C3: set_real,B: real,D3: set_real] :
( ( member_real @ A3 @ C3 )
=> ( ( member_real @ B @ D3 )
=> ( member_real @ ( times_times_real @ A3 @ B ) @ ( times_times_set_real @ C3 @ D3 ) ) ) ) ).
% set_times_intro
thf(fact_357_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_358_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_359_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_360_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_361_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_362_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_363_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_364_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_365_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_366_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_367_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_368_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_369_set__times__elim,axiom,
! [X: mat_a,A: set_mat_a,B4: set_mat_a] :
( ( member_mat_a @ X @ ( times_1230744552615602198_mat_a @ A @ B4 ) )
=> ~ ! [A2: mat_a,B3: mat_a] :
( ( X
= ( times_times_mat_a @ A2 @ B3 ) )
=> ( ( member_mat_a @ A2 @ A )
=> ~ ( member_mat_a @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_370_set__times__elim,axiom,
! [X: a,A: set_a,B4: set_a] :
( ( member_a @ X @ ( times_times_set_a @ A @ B4 ) )
=> ~ ! [A2: a,B3: a] :
( ( X
= ( times_times_a @ A2 @ B3 ) )
=> ( ( member_a @ A2 @ A )
=> ~ ( member_a @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_371_set__times__elim,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A @ B4 ) )
=> ~ ! [A2: nat,B3: nat] :
( ( X
= ( times_times_nat @ A2 @ B3 ) )
=> ( ( member_nat @ A2 @ A )
=> ~ ( member_nat @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_372_set__times__elim,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A @ B4 ) )
=> ~ ! [A2: int,B3: int] :
( ( X
= ( times_times_int @ A2 @ B3 ) )
=> ( ( member_int @ A2 @ A )
=> ~ ( member_int @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_373_set__times__elim,axiom,
! [X: real,A: set_real,B4: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A @ B4 ) )
=> ~ ! [A2: real,B3: real] :
( ( X
= ( times_times_real @ A2 @ B3 ) )
=> ( ( member_real @ A2 @ A )
=> ~ ( member_real @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_374_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_375_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_376_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_377_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_378_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_379_in__mono,axiom,
! [A: set_vec_a,B4: set_vec_a,X: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B4 )
=> ( ( member_vec_a @ X @ A )
=> ( member_vec_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_380_in__mono,axiom,
! [A: set_mat_a,B4: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B4 )
=> ( ( member_mat_a @ X @ A )
=> ( member_mat_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_381_in__mono,axiom,
! [A: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_382_in__mono,axiom,
! [A: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_383_subsetD,axiom,
! [A: set_vec_a,B4: set_vec_a,C: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B4 )
=> ( ( member_vec_a @ C @ A )
=> ( member_vec_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_384_subsetD,axiom,
! [A: set_mat_a,B4: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B4 )
=> ( ( member_mat_a @ C @ A )
=> ( member_mat_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_385_subsetD,axiom,
! [A: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_386_subsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_387_equalityE,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% equalityE
thf(fact_388_subset__eq,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A6: set_vec_a,B6: set_vec_a] :
! [X4: vec_a] :
( ( member_vec_a @ X4 @ A6 )
=> ( member_vec_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_389_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A6: set_mat_a,B6: set_mat_a] :
! [X4: mat_a] :
( ( member_mat_a @ X4 @ A6 )
=> ( member_mat_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_390_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( member_real @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_391_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_392_equalityD1,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_393_Set_OequalityD2,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% Set.equalityD2
thf(fact_394_subset__iff,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A6: set_vec_a,B6: set_vec_a] :
! [T2: vec_a] :
( ( member_vec_a @ T2 @ A6 )
=> ( member_vec_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_395_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A6: set_mat_a,B6: set_mat_a] :
! [T2: mat_a] :
( ( member_mat_a @ T2 @ A6 )
=> ( member_mat_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_396_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_398_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_399_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_400_subset__trans,axiom,
! [A: set_nat,B4: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_401_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_402_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_403_Collect__subset,axiom,
! [A: set_vec_a,P: vec_a > $o] :
( ord_le4791951621262958845_vec_a
@ ( collect_vec_a
@ ^ [X4: vec_a] :
( ( member_vec_a @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_404_Collect__subset,axiom,
! [A: set_mat_a,P: mat_a > $o] :
( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X4: mat_a] :
( ( member_mat_a @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_405_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_406_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_407_less__eq__set__def,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A6: set_vec_a,B6: set_vec_a] :
( ord_less_eq_vec_a_o
@ ^ [X4: vec_a] : ( member_vec_a @ X4 @ A6 )
@ ^ [X4: vec_a] : ( member_vec_a @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_408_less__eq__set__def,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A6: set_mat_a,B6: set_mat_a] :
( ord_less_eq_mat_a_o
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ A6 )
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_409_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A6 )
@ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_410_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_411_norm1__ge__0,axiom,
! [F: poly_real] : ( ord_less_eq_real @ zero_zero_real @ ( norm1_real @ F ) ) ).
% norm1_ge_0
thf(fact_412_norm1__ge__0,axiom,
! [F: poly_int] : ( ord_less_eq_int @ zero_zero_int @ ( norm1_int @ F ) ) ).
% norm1_ge_0
thf(fact_413_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_414_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_415_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_416_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_417_dual__order_Orefl,axiom,
! [A3: a] : ( ord_less_eq_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_418_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_419_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_420_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_421_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_422_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_423_mat__diag__diag,axiom,
! [N: nat,F: nat > a,G: nat > a] :
( ( times_times_mat_a @ ( mat_diag_a @ N @ F ) @ ( mat_diag_a @ N @ G ) )
= ( mat_diag_a @ N
@ ^ [I: nat] : ( times_times_a @ ( F @ I ) @ ( G @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_424_mat__diag__diag,axiom,
! [N: nat,F: nat > nat,G: nat > nat] :
( ( times_times_mat_nat @ ( mat_diag_nat @ N @ F ) @ ( mat_diag_nat @ N @ G ) )
= ( mat_diag_nat @ N
@ ^ [I: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_425_mat__diag__diag,axiom,
! [N: nat,F: nat > int,G: nat > int] :
( ( times_times_mat_int @ ( mat_diag_int @ N @ F ) @ ( mat_diag_int @ N @ G ) )
= ( mat_diag_int @ N
@ ^ [I: nat] : ( times_times_int @ ( F @ I ) @ ( G @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_426_mat__diag__diag,axiom,
! [N: nat,F: nat > real,G: nat > real] :
( ( times_times_mat_real @ ( mat_diag_real @ N @ F ) @ ( mat_diag_real @ N @ G ) )
= ( mat_diag_real @ N
@ ^ [I: nat] : ( times_times_real @ ( F @ I ) @ ( G @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_427_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_428_linf__norm__vec__eq__0,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( ( linf_norm_vec_int @ V )
= zero_zero_int )
= ( V
= ( zero_vec_int @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_429_linf__norm__vec__eq__0,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( ( linf_norm_vec_real @ V )
= zero_zero_real )
= ( V
= ( zero_vec_real @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_430_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_431_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_432_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_433_index__update__vec2,axiom,
! [I4: nat,I2: nat,V: vec_a,A3: a] :
( ( I4 != I2 )
=> ( ( vec_index_a @ ( update_vec_a @ V @ I2 @ A3 ) @ I4 )
= ( vec_index_a @ V @ I4 ) ) ) ).
% index_update_vec2
thf(fact_434_dim__update__vec,axiom,
! [V: vec_a,I2: nat,A3: a] :
( ( dim_vec_a @ ( update_vec_a @ V @ I2 @ A3 ) )
= ( dim_vec_a @ V ) ) ).
% dim_update_vec
thf(fact_435_linf__norm__vec__ge__0,axiom,
! [V: vec_real] : ( ord_less_eq_real @ zero_zero_real @ ( linf_norm_vec_real @ V ) ) ).
% linf_norm_vec_ge_0
thf(fact_436_linf__norm__vec__ge__0,axiom,
! [V: vec_int] : ( ord_less_eq_int @ zero_zero_int @ ( linf_norm_vec_int @ V ) ) ).
% linf_norm_vec_ge_0
thf(fact_437_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_int @ ( zero_vec_int @ N ) )
= zero_zero_int ) ).
% linf_norm_zero_vec
thf(fact_438_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_real @ ( zero_vec_real @ N ) )
= zero_zero_real ) ).
% linf_norm_zero_vec
thf(fact_439_mat__diag__dim,axiom,
! [N: nat,F: nat > a] : ( member_mat_a @ ( mat_diag_a @ N @ F ) @ ( carrier_mat_a @ N @ N ) ) ).
% mat_diag_dim
thf(fact_440_nle__le,axiom,
! [A3: real,B: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B ) )
= ( ( ord_less_eq_real @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_441_nle__le,axiom,
! [A3: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B ) )
= ( ( ord_less_eq_nat @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_442_nle__le,axiom,
! [A3: int,B: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B ) )
= ( ( ord_less_eq_int @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_443_nle__le,axiom,
! [A3: a,B: a] :
( ( ~ ( ord_less_eq_a @ A3 @ B ) )
= ( ( ord_less_eq_a @ B @ A3 )
& ( B != A3 ) ) ) ).
% nle_le
thf(fact_444_le__cases3,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_445_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_446_le__cases3,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_447_le__cases3,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ( ord_less_eq_a @ X @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X )
=> ~ ( ord_less_eq_a @ X @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X )
=> ~ ( ord_less_eq_a @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_448_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_449_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_450_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_451_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_452_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z: a] : ( Y5 = Z ) )
= ( ^ [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_453_ord__eq__le__trans,axiom,
! [A3: real,B: real,C: real] :
( ( A3 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_454_ord__eq__le__trans,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( A3 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_455_ord__eq__le__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( A3 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_456_ord__eq__le__trans,axiom,
! [A3: int,B: int,C: int] :
( ( A3 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_457_ord__eq__le__trans,axiom,
! [A3: a,B: a,C: a] :
( ( A3 = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_458_ord__le__eq__trans,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_459_ord__le__eq__trans,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_460_ord__le__eq__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_461_ord__le__eq__trans,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_462_ord__le__eq__trans,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_463_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_464_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_465_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_466_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_467_order__antisym,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_468_order_Otrans,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% order.trans
thf(fact_469_order_Otrans,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_470_order_Otrans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_471_order_Otrans,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A3 @ C ) ) ) ).
% order.trans
thf(fact_472_order_Otrans,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A3 @ C ) ) ) ).
% order.trans
thf(fact_473_order__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_474_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_475_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_476_order__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_477_order__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_478_linorder__wlog,axiom,
! [P: real > real > $o,A3: real,B: real] :
( ! [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: real,B3: real] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_479_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: nat,B3: nat] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_480_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B: int] :
( ! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: int,B3: int] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_481_linorder__wlog,axiom,
! [P: a > a > $o,A3: a,B: a] :
( ! [A2: a,B3: a] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: a,B3: a] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ).
% linorder_wlog
thf(fact_482_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_483_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_484_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_485_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_486_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z: a] : ( Y5 = Z ) )
= ( ^ [A4: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A4 )
& ( ord_less_eq_a @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_487_dual__order_Oantisym,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_488_dual__order_Oantisym,axiom,
! [B: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_489_dual__order_Oantisym,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_490_dual__order_Oantisym,axiom,
! [B: int,A3: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_491_dual__order_Oantisym,axiom,
! [B: a,A3: a] :
( ( ord_less_eq_a @ B @ A3 )
=> ( ( ord_less_eq_a @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_492_dual__order_Otrans,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_493_dual__order_Otrans,axiom,
! [B: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_494_dual__order_Otrans,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_495_dual__order_Otrans,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_496_dual__order_Otrans,axiom,
! [B: a,A3: a,C: a] :
( ( ord_less_eq_a @ B @ A3 )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_497_antisym,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_498_antisym,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_499_antisym,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_500_antisym,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_501_antisym,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_eq_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_502_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_503_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_504_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_505_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_506_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z: a] : ( Y5 = Z ) )
= ( ^ [A4: a,B2: a] :
( ( ord_less_eq_a @ A4 @ B2 )
& ( ord_less_eq_a @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_507_order__subst1,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_509_order__subst1,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_510_order__subst1,axiom,
! [A3: real,F: a > real,B: a,C: a] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_511_order__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_512_order__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_513_order__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_514_order__subst1,axiom,
! [A3: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_515_order__subst1,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_516_order__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_517_order__subst2,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_518_order__subst2,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_519_order__subst2,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_520_order__subst2,axiom,
! [A3: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_521_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_522_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_523_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_524_order__subst2,axiom,
! [A3: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_525_order__subst2,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_526_order__subst2,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_527_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_528_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_529_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_530_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_531_order__eq__refl,axiom,
! [X: a,Y2: a] :
( ( X = Y2 )
=> ( ord_less_eq_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_532_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_533_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_534_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_535_linorder__linear,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
| ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_536_ord__eq__le__subst,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_537_ord__eq__le__subst,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_538_ord__eq__le__subst,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_539_ord__eq__le__subst,axiom,
! [A3: a,F: real > a,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_540_ord__eq__le__subst,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_541_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_542_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_543_ord__eq__le__subst,axiom,
! [A3: a,F: nat > a,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_544_ord__eq__le__subst,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_545_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_546_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_547_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_548_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_549_ord__le__eq__subst,axiom,
! [A3: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_550_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_551_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_552_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_553_ord__le__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_554_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_555_ord__le__eq__subst,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_556_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_557_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_558_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_559_linorder__le__cases,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_560_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_561_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_562_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_563_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_564_order__antisym__conv,axiom,
! [Y2: a,X: a] :
( ( ord_less_eq_a @ Y2 @ X )
=> ( ( ord_less_eq_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_565_pred__subset__eq,axiom,
! [R2: set_vec_a,S: set_vec_a] :
( ( ord_less_eq_vec_a_o
@ ^ [X4: vec_a] : ( member_vec_a @ X4 @ R2 )
@ ^ [X4: vec_a] : ( member_vec_a @ X4 @ S ) )
= ( ord_le4791951621262958845_vec_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_566_pred__subset__eq,axiom,
! [R2: set_mat_a,S: set_mat_a] :
( ( ord_less_eq_mat_a_o
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ R2 )
@ ^ [X4: mat_a] : ( member_mat_a @ X4 @ S ) )
= ( ord_le3318621148231462513_mat_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_567_pred__subset__eq,axiom,
! [R2: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ R2 )
@ ^ [X4: real] : ( member_real @ X4 @ S ) )
= ( ord_less_eq_set_real @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_568_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_569_linf__norm__vec__greater__0,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( ord_less_int @ zero_zero_int @ ( linf_norm_vec_int @ V ) )
= ( V
!= ( zero_vec_int @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_570_linf__norm__vec__greater__0,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( ord_less_real @ zero_zero_real @ ( linf_norm_vec_real @ V ) )
= ( V
!= ( zero_vec_real @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_571_index__update__vec1,axiom,
! [I2: nat,V: vec_a,A3: a] :
( ( ord_less_nat @ I2 @ ( dim_vec_a @ V ) )
=> ( ( vec_index_a @ ( update_vec_a @ V @ I2 @ A3 ) @ I2 )
= A3 ) ) ).
% index_update_vec1
thf(fact_572_sq__norm__poly__ge__0,axiom,
! [P2: poly_real] : ( ord_less_eq_real @ zero_zero_real @ ( sq_norm_poly_real @ P2 ) ) ).
% sq_norm_poly_ge_0
thf(fact_573_sq__norm__poly__ge__0,axiom,
! [P2: poly_int] : ( ord_less_eq_int @ zero_zero_int @ ( sq_norm_poly_int @ P2 ) ) ).
% sq_norm_poly_ge_0
thf(fact_574_less__eq__vec__def,axiom,
( ord_less_eq_vec_real
= ( ^ [V2: vec_real,W: vec_real] :
( ( ( dim_vec_real @ V2 )
= ( dim_vec_real @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_real @ W ) )
=> ( ord_less_eq_real @ ( vec_index_real @ V2 @ I ) @ ( vec_index_real @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_575_less__eq__vec__def,axiom,
( ord_le2754941374173645459et_nat
= ( ^ [V2: vec_set_nat,W: vec_set_nat] :
( ( ( dim_vec_set_nat @ V2 )
= ( dim_vec_set_nat @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_set_nat @ W ) )
=> ( ord_less_eq_set_nat @ ( vec_index_set_nat @ V2 @ I ) @ ( vec_index_set_nat @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_576_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V2: vec_nat,W: vec_nat] :
( ( ( dim_vec_nat @ V2 )
= ( dim_vec_nat @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ W ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V2 @ I ) @ ( vec_index_nat @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_577_less__eq__vec__def,axiom,
( ord_less_eq_vec_int
= ( ^ [V2: vec_int,W: vec_int] :
( ( ( dim_vec_int @ V2 )
= ( dim_vec_int @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_int @ W ) )
=> ( ord_less_eq_int @ ( vec_index_int @ V2 @ I ) @ ( vec_index_int @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_578_less__eq__vec__def,axiom,
( ord_less_eq_vec_a
= ( ^ [V2: vec_a,W: vec_a] :
( ( ( dim_vec_a @ V2 )
= ( dim_vec_a @ W ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_a @ W ) )
=> ( ord_less_eq_a @ ( vec_index_a @ V2 @ I ) @ ( vec_index_a @ W @ I ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_579_conjugate__square__eq__0__vec,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( ( scalar_prod_int @ V @ ( conjug4884081946748880444ec_int @ V ) )
= zero_zero_int )
= ( V
= ( zero_vec_int @ N ) ) ) ) ).
% conjugate_square_eq_0_vec
thf(fact_580_conjugate__square__eq__0__vec,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( ( scalar_prod_real @ V @ ( conjug3612589096773959740c_real @ V ) )
= zero_zero_real )
= ( V
= ( zero_vec_real @ N ) ) ) ) ).
% conjugate_square_eq_0_vec
thf(fact_581_subset__Collect__iff,axiom,
! [B4: set_vec_a,A: set_vec_a,P: vec_a > $o] :
( ( ord_le4791951621262958845_vec_a @ B4 @ A )
=> ( ( ord_le4791951621262958845_vec_a @ B4
@ ( collect_vec_a
@ ^ [X4: vec_a] :
( ( member_vec_a @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: vec_a] :
( ( member_vec_a @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_582_subset__Collect__iff,axiom,
! [B4: set_mat_a,A: set_mat_a,P: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ B4 @ A )
=> ( ( ord_le3318621148231462513_mat_a @ B4
@ ( collect_mat_a
@ ^ [X4: mat_a] :
( ( member_mat_a @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: mat_a] :
( ( member_mat_a @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_583_subset__Collect__iff,axiom,
! [B4: set_real,A: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B4 @ A )
=> ( ( ord_less_eq_set_real @ B4
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: real] :
( ( member_real @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_584_subset__Collect__iff,axiom,
! [B4: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_585_subset__CollectI,axiom,
! [B4: set_vec_a,A: set_vec_a,Q: vec_a > $o,P: vec_a > $o] :
( ( ord_le4791951621262958845_vec_a @ B4 @ A )
=> ( ! [X2: vec_a] :
( ( member_vec_a @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le4791951621262958845_vec_a
@ ( collect_vec_a
@ ^ [X4: vec_a] :
( ( member_vec_a @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_vec_a
@ ^ [X4: vec_a] :
( ( member_vec_a @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_586_subset__CollectI,axiom,
! [B4: set_mat_a,A: set_mat_a,Q: mat_a > $o,P: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ B4 @ A )
=> ( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X4: mat_a] :
( ( member_mat_a @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_mat_a
@ ^ [X4: mat_a] :
( ( member_mat_a @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_587_subset__CollectI,axiom,
! [B4: set_real,A: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B4 @ A )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_588_subset__CollectI,axiom,
! [B4: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_589_Collect__restrict,axiom,
! [X5: set_vec_a,P: vec_a > $o] :
( ord_le4791951621262958845_vec_a
@ ( collect_vec_a
@ ^ [X4: vec_a] :
( ( member_vec_a @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_590_Collect__restrict,axiom,
! [X5: set_mat_a,P: mat_a > $o] :
( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X4: mat_a] :
( ( member_mat_a @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_591_Collect__restrict,axiom,
! [X5: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_592_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_593_psubsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_nat @ A @ B4 ) ) ) ).
% psubsetI
thf(fact_594_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_595_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_596_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_597_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_598_sq__norm__poly__eq__0,axiom,
! [P2: poly_int] :
( ( ( sq_norm_poly_int @ P2 )
= zero_zero_int )
= ( P2 = zero_zero_poly_int ) ) ).
% sq_norm_poly_eq_0
thf(fact_599_sq__norm__poly__eq__0,axiom,
! [P2: poly_real] :
( ( ( sq_norm_poly_real @ P2 )
= zero_zero_real )
= ( P2 = zero_zero_poly_real ) ) ).
% sq_norm_poly_eq_0
thf(fact_600_sq__norm__poly__0,axiom,
( ( sq_norm_poly_int @ zero_zero_poly_int )
= zero_zero_int ) ).
% sq_norm_poly_0
thf(fact_601_sq__norm__poly__0,axiom,
( ( sq_norm_poly_real @ zero_zero_poly_real )
= zero_zero_real ) ).
% sq_norm_poly_0
thf(fact_602_atLeastLessThan__iff,axiom,
! [I2: vec_a,L: vec_a,U: vec_a] :
( ( member_vec_a @ I2 @ ( set_or2357829874413910678_vec_a @ L @ U ) )
= ( ( ord_less_eq_vec_a @ L @ I2 )
& ( ord_less_vec_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_603_atLeastLessThan__iff,axiom,
! [I2: mat_a,L: mat_a,U: mat_a] :
( ( member_mat_a @ I2 @ ( set_or1377778852321182218_mat_a @ L @ U ) )
= ( ( ord_less_eq_mat_a @ L @ I2 )
& ( ord_less_mat_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_604_atLeastLessThan__iff,axiom,
! [I2: real,L: real,U: real] :
( ( member_real @ I2 @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_real @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_605_atLeastLessThan__iff,axiom,
! [I2: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_set_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_606_atLeastLessThan__iff,axiom,
! [I2: int,L: int,U: int] :
( ( member_int @ I2 @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I2 )
& ( ord_less_int @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_607_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_608_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_609_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_610_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_611_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_612_eq__vecI,axiom,
! [W2: vec_a,V: vec_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_a @ W2 ) )
=> ( ( vec_index_a @ V @ I3 )
= ( vec_index_a @ W2 @ I3 ) ) )
=> ( ( ( dim_vec_a @ V )
= ( dim_vec_a @ W2 ) )
=> ( V = W2 ) ) ) ).
% eq_vecI
thf(fact_613_sq__norm__poly__pos,axiom,
! [P2: poly_int] :
( ( ord_less_int @ zero_zero_int @ ( sq_norm_poly_int @ P2 ) )
= ( P2 != zero_zero_poly_int ) ) ).
% sq_norm_poly_pos
thf(fact_614_sq__norm__poly__pos,axiom,
! [P2: poly_real] :
( ( ord_less_real @ zero_zero_real @ ( sq_norm_poly_real @ P2 ) )
= ( P2 != zero_zero_poly_real ) ) ).
% sq_norm_poly_pos
thf(fact_615_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_616_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_617_index__zero__vec_I1_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_nat @ ( zero_vec_nat @ N ) @ I2 )
= zero_zero_nat ) ) ).
% index_zero_vec(1)
thf(fact_618_index__zero__vec_I1_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_int @ ( zero_vec_int @ N ) @ I2 )
= zero_zero_int ) ) ).
% index_zero_vec(1)
thf(fact_619_index__zero__vec_I1_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_real @ ( zero_vec_real @ N ) @ I2 )
= zero_zero_real ) ) ).
% index_zero_vec(1)
thf(fact_620_index__zero__vec_I1_J,axiom,
! [I2: nat,N: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( vec_index_a @ ( zero_vec_a @ N ) @ I2 )
= zero_zero_a ) ) ).
% index_zero_vec(1)
thf(fact_621_conjugate__square__ge__0__vec,axiom,
! [V: vec_real] : ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ V @ ( conjug3612589096773959740c_real @ V ) ) ) ).
% conjugate_square_ge_0_vec
thf(fact_622_conjugate__square__ge__0__vec,axiom,
! [V: vec_int] : ( ord_less_eq_int @ zero_zero_int @ ( scalar_prod_int @ V @ ( conjug4884081946748880444ec_int @ V ) ) ) ).
% conjugate_square_ge_0_vec
thf(fact_623_conjugate__square__greater__0__vec,axiom,
! [V: vec_int,N: nat] :
( ( member_vec_int @ V @ ( carrier_vec_int @ N ) )
=> ( ( ord_less_int @ zero_zero_int @ ( scalar_prod_int @ V @ ( conjug4884081946748880444ec_int @ V ) ) )
= ( V
!= ( zero_vec_int @ N ) ) ) ) ).
% conjugate_square_greater_0_vec
thf(fact_624_conjugate__square__greater__0__vec,axiom,
! [V: vec_real,N: nat] :
( ( member_vec_real @ V @ ( carrier_vec_real @ N ) )
=> ( ( ord_less_real @ zero_zero_real @ ( scalar_prod_real @ V @ ( conjug3612589096773959740c_real @ V ) ) )
= ( V
!= ( zero_vec_real @ N ) ) ) ) ).
% conjugate_square_greater_0_vec
thf(fact_625_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_626_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_627_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_628_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_629_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_630_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_631_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_632_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_633_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_634_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_635_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_636_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_637_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_638_order__less__imp__triv,axiom,
! [X: int,Y2: int,P: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_639_order__less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_640_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_641_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_642_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_643_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_644_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_645_order__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_646_order__less__subst2,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_647_order__less__subst2,axiom,
! [A3: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_648_order__less__subst2,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_649_order__less__subst2,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_650_order__less__subst2,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_651_order__less__subst2,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_652_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_653_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_654_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_655_order__less__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_656_order__less__subst1,axiom,
! [A3: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_657_order__less__subst1,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_658_order__less__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_659_order__less__subst1,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_660_order__less__subst1,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_661_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_662_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_663_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_664_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_665_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_666_ord__less__eq__subst,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_667_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_668_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_669_ord__less__eq__subst,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_670_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_671_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_672_ord__less__eq__subst,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_673_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_674_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_675_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_676_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_677_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B: int,C: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_678_ord__eq__less__subst,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_679_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_680_ord__eq__less__subst,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_681_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_682_order__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_683_order__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_684_order__less__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_685_order__less__asym_H,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ~ ( ord_less_nat @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_686_order__less__asym_H,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ~ ( ord_less_int @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_687_order__less__asym_H,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ~ ( ord_less_real @ B @ A3 ) ) ).
% order_less_asym'
thf(fact_688_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_689_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_690_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_691_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_692_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_693_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_694_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_695_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_696_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_697_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_698_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_699_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ( A3 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_700_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_701_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_702_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( A3 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_703_dual__order_Ostrict__trans,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_704_dual__order_Ostrict__trans,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_705_dual__order_Ostrict__trans,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_706_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_707_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_708_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_709_order_Ostrict__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_710_order_Ostrict__trans,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_711_order_Ostrict__trans,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_712_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B3: nat] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_713_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B: int] :
( ! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B3: int] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_714_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B: real] :
( ! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( P @ A2 @ B3 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B3: real] :
( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A3 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_715_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P4 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_716_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_717_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_718_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_719_dual__order_Oasym,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ~ ( ord_less_nat @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_720_dual__order_Oasym,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ~ ( ord_less_int @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_721_dual__order_Oasym,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ~ ( ord_less_real @ A3 @ B ) ) ).
% dual_order.asym
thf(fact_722_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_723_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_724_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_725_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_726_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_727_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_728_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_729_ord__less__eq__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_730_ord__less__eq__trans,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_731_ord__less__eq__trans,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_732_ord__eq__less__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( A3 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_733_ord__eq__less__trans,axiom,
! [A3: int,B: int,C: int] :
( ( A3 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_734_ord__eq__less__trans,axiom,
! [A3: real,B: real,C: real] :
( ( A3 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_735_order_Oasym,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ~ ( ord_less_nat @ B @ A3 ) ) ).
% order.asym
thf(fact_736_order_Oasym,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ~ ( ord_less_int @ B @ A3 ) ) ).
% order.asym
thf(fact_737_order_Oasym,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ~ ( ord_less_real @ B @ A3 ) ) ).
% order.asym
thf(fact_738_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_739_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_740_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_741_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_742_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_743_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_744_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_745_lt__ex,axiom,
! [X: int] :
? [Y: int] : ( ord_less_int @ Y @ X ) ).
% lt_ex
thf(fact_746_lt__ex,axiom,
! [X: real] :
? [Y: real] : ( ord_less_real @ Y @ X ) ).
% lt_ex
thf(fact_747_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_748_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_749_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_750_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_751_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_752_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_753_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_754_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_755_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_756_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_757_real__linorder__cases,axiom,
! [A3: nat,B: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ~ ( ord_less_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_nat @ B @ A3 ) ) ) ) ) ).
% real_linorder_cases
thf(fact_758_real__linorder__cases,axiom,
! [A3: int,B: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ~ ( ord_less_int @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_int @ B @ A3 ) ) ) ) ) ).
% real_linorder_cases
thf(fact_759_real__linorder__cases,axiom,
! [A3: real,B: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ~ ( ord_less_real @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_real @ B @ A3 ) ) ) ) ) ).
% real_linorder_cases
thf(fact_760_pos__pos__linear,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ A3 @ B )
| ( A3 = B )
| ( ord_less_nat @ B @ A3 ) ) ) ) ).
% pos_pos_linear
thf(fact_761_pos__pos__linear,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ A3 @ B )
| ( A3 = B )
| ( ord_less_int @ B @ A3 ) ) ) ) ).
% pos_pos_linear
thf(fact_762_pos__pos__linear,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ A3 @ B )
| ( A3 = B )
| ( ord_less_real @ B @ A3 ) ) ) ) ).
% pos_pos_linear
thf(fact_763_neg__neg__linear,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ A3 @ B )
| ( A3 = B )
| ( ord_less_nat @ B @ A3 ) ) ) ) ).
% neg_neg_linear
thf(fact_764_neg__neg__linear,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ A3 @ B )
| ( A3 = B )
| ( ord_less_int @ B @ A3 ) ) ) ) ).
% neg_neg_linear
thf(fact_765_neg__neg__linear,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ A3 @ B )
| ( A3 = B )
| ( ord_less_real @ B @ A3 ) ) ) ) ).
% neg_neg_linear
thf(fact_766_real__linear,axiom,
! [A3: nat,B: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ A3 @ B )
| ( A3 = B )
| ( ord_less_nat @ B @ A3 ) ) ) ) ).
% real_linear
thf(fact_767_real__linear,axiom,
! [A3: int,B: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ A3 @ B )
| ( A3 = B )
| ( ord_less_int @ B @ A3 ) ) ) ) ).
% real_linear
thf(fact_768_real__linear,axiom,
! [A3: real,B: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ A3 @ B )
| ( A3 = B )
| ( ord_less_real @ B @ A3 ) ) ) ) ).
% real_linear
thf(fact_769_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_770_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_771_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_772_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_773_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_774_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_775_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_776_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_777_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_778_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_779_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_780_order__le__imp__less__or__eq,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_781_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_782_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_783_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_784_linorder__le__less__linear,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
| ( ord_less_a @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_785_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_786_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_787_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_788_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_789_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_790_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_791_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_792_order__less__le__subst2,axiom,
! [A3: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_793_order__less__le__subst2,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_794_order__less__le__subst2,axiom,
! [A3: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_795_order__less__le__subst1,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_796_order__less__le__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_797_order__less__le__subst1,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_798_order__less__le__subst1,axiom,
! [A3: a,F: real > a,B: real,C: real] :
( ( ord_less_a @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_799_order__less__le__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_800_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_801_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_802_order__less__le__subst1,axiom,
! [A3: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_803_order__less__le__subst1,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_804_order__less__le__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_805_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_806_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_807_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_808_order__le__less__subst2,axiom,
! [A3: real,B: real,F: real > a,C: a] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_809_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_810_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_811_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_812_order__le__less__subst2,axiom,
! [A3: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_813_order__le__less__subst2,axiom,
! [A3: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_814_order__le__less__subst2,axiom,
! [A3: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_815_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_816_order__le__less__subst1,axiom,
! [A3: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_817_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_818_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_819_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_820_order__le__less__subst1,axiom,
! [A3: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_821_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_822_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_823_order__le__less__subst1,axiom,
! [A3: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_824_order__le__less__subst1,axiom,
! [A3: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A3 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_825_order__less__le__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_826_order__less__le__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_827_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_828_order__less__le__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_829_order__less__le__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_830_order__le__less__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_831_order__le__less__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_832_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_833_order__le__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_834_order__le__less__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_835_order__neq__le__trans,axiom,
! [A3: real,B: real] :
( ( A3 != B )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_836_order__neq__le__trans,axiom,
! [A3: set_nat,B: set_nat] :
( ( A3 != B )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ord_less_set_nat @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_837_order__neq__le__trans,axiom,
! [A3: nat,B: nat] :
( ( A3 != B )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_838_order__neq__le__trans,axiom,
! [A3: int,B: int] :
( ( A3 != B )
=> ( ( ord_less_eq_int @ A3 @ B )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_839_order__neq__le__trans,axiom,
! [A3: a,B: a] :
( ( A3 != B )
=> ( ( ord_less_eq_a @ A3 @ B )
=> ( ord_less_a @ A3 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_840_order__le__neq__trans,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_841_order__le__neq__trans,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_set_nat @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_842_order__le__neq__trans,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_843_order__le__neq__trans,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_844_order__le__neq__trans,axiom,
! [A3: a,B: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_a @ A3 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_845_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_846_order__less__imp__le,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_847_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_848_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_849_order__less__imp__le,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_eq_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_850_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_851_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_852_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_853_linorder__not__less,axiom,
! [X: a,Y2: a] :
( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_854_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_855_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_856_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_857_linorder__not__le,axiom,
! [X: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X @ Y2 ) )
= ( ord_less_a @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_858_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_859_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_860_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_861_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_862_order__less__le,axiom,
( ord_less_a
= ( ^ [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_863_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_864_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_865_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_866_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_867_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_868_dual__order_Ostrict__implies__order,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ B @ A3 )
=> ( ord_less_eq_real @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_869_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B @ A3 )
=> ( ord_less_eq_set_nat @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_870_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ord_less_eq_nat @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_871_dual__order_Ostrict__implies__order,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ B @ A3 )
=> ( ord_less_eq_int @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_872_dual__order_Ostrict__implies__order,axiom,
! [B: a,A3: a] :
( ( ord_less_a @ B @ A3 )
=> ( ord_less_eq_a @ B @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_873_order_Ostrict__implies__order,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ B )
=> ( ord_less_eq_real @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_874_order_Ostrict__implies__order,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A3 @ B )
=> ( ord_less_eq_set_nat @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_875_order_Ostrict__implies__order,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ord_less_eq_nat @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_876_order_Ostrict__implies__order,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ B )
=> ( ord_less_eq_int @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_877_order_Ostrict__implies__order,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ A3 @ B )
=> ( ord_less_eq_a @ A3 @ B ) ) ).
% order.strict_implies_order
thf(fact_878_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_879_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_880_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_881_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_882_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A4: a] :
( ( ord_less_eq_a @ B2 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_883_dual__order_Ostrict__trans2,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_884_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_885_dual__order_Ostrict__trans2,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B @ A3 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_886_dual__order_Ostrict__trans2,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_887_dual__order_Ostrict__trans2,axiom,
! [B: a,A3: a,C: a] :
( ( ord_less_a @ B @ A3 )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_888_dual__order_Ostrict__trans1,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_889_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_890_dual__order_Ostrict__trans1,axiom,
! [B: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_891_dual__order_Ostrict__trans1,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_eq_int @ B @ A3 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_892_dual__order_Ostrict__trans1,axiom,
! [B: a,A3: a,C: a] :
( ( ord_less_eq_a @ B @ A3 )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_893_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_894_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_895_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_896_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_897_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A4: a] :
( ( ord_less_eq_a @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_898_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_899_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_900_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_901_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_902_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A4: a] :
( ( ord_less_a @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_903_dense__le__bounded,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W3: real] :
( ( ord_less_real @ X @ W3 )
=> ( ( ord_less_real @ W3 @ Y2 )
=> ( ord_less_eq_real @ W3 @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_904_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y2: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W3: real] :
( ( ord_less_real @ Z2 @ W3 )
=> ( ( ord_less_real @ W3 @ X )
=> ( ord_less_eq_real @ Y2 @ W3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_905_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_906_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_907_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_908_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_909_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B2: a] :
( ( ord_less_eq_a @ A4 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_910_order_Ostrict__trans2,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_911_order_Ostrict__trans2,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A3 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_912_order_Ostrict__trans2,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_913_order_Ostrict__trans2,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_914_order_Ostrict__trans2,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_915_order_Ostrict__trans1,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_916_order_Ostrict__trans1,axiom,
! [A3: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_917_order_Ostrict__trans1,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_918_order_Ostrict__trans1,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_919_order_Ostrict__trans1,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_920_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_921_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_922_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_923_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_924_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B2: a] :
( ( ord_less_eq_a @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_925_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_926_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_927_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_928_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_929_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B2: a] :
( ( ord_less_a @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_930_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_931_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_932_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_933_not__le__imp__less,axiom,
! [Y2: a,X: a] :
( ~ ( ord_less_eq_a @ Y2 @ X )
=> ( ord_less_a @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_934_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_935_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_936_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_937_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_938_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
& ~ ( ord_less_eq_a @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_939_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_940_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_941_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_942_antisym__conv2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_943_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_944_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_945_antisym__conv2,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_946_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_947_antisym__conv1,axiom,
! [X: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_948_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_949_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_950_antisym__conv1,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_951_nless__le,axiom,
! [A3: real,B: real] :
( ( ~ ( ord_less_real @ A3 @ B ) )
= ( ~ ( ord_less_eq_real @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_952_nless__le,axiom,
! [A3: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A3 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_953_nless__le,axiom,
! [A3: nat,B: nat] :
( ( ~ ( ord_less_nat @ A3 @ B ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_954_nless__le,axiom,
! [A3: int,B: int] :
( ( ~ ( ord_less_int @ A3 @ B ) )
= ( ~ ( ord_less_eq_int @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_955_nless__le,axiom,
! [A3: a,B: a] :
( ( ~ ( ord_less_a @ A3 @ B ) )
= ( ~ ( ord_less_eq_a @ A3 @ B )
| ( A3 = B ) ) ) ).
% nless_le
thf(fact_956_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_957_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_958_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_959_leI,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_a @ X @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) ) ).
% leI
thf(fact_960_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_961_leD,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ~ ( ord_less_set_nat @ X @ Y2 ) ) ).
% leD
thf(fact_962_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_963_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_964_leD,axiom,
! [Y2: a,X: a] :
( ( ord_less_eq_a @ Y2 @ X )
=> ~ ( ord_less_a @ X @ Y2 ) ) ).
% leD
thf(fact_965_psubsetE,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% psubsetE
thf(fact_966_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_967_psubset__imp__subset,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% psubset_imp_subset
thf(fact_968_psubset__subset__trans,axiom,
! [A: set_nat,B4: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_969_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_970_subset__psubset__trans,axiom,
! [A: set_nat,B4: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_971_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_972_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_973_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_974_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_975_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_976_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_977_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_978_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_979_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_980_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_981_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_982_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_983_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_984_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_985_atLeastLessThan__eq__iff,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A3 @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A3 = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_986_atLeastLessThan__eq__iff,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A3 @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A3 = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_987_atLeastLessThan__eq__iff,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A3 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A3 = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_988_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H3: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_989_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H3: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_990_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H = H3 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_991_atLeastLessThan__inj_I1_J,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A3 @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A3 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_992_atLeastLessThan__inj_I1_J,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A3 @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A3 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_993_atLeastLessThan__inj_I1_J,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A3 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A3 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_994_atLeastLessThan__inj_I2_J,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A3 @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_995_atLeastLessThan__inj_I2_J,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A3 @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_996_atLeastLessThan__inj_I2_J,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A3 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_997_nonpos__linorder__cases,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ~ ( ord_less_real @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_real @ B @ A3 ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_998_nonpos__linorder__cases,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ~ ( ord_less_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_nat @ B @ A3 ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_999_nonpos__linorder__cases,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ~ ( ord_less_int @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_int @ B @ A3 ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_1000_nonneg__linorder__cases,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ~ ( ord_less_real @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_real @ B @ A3 ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_1001_nonneg__linorder__cases,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ~ ( ord_less_nat @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_nat @ B @ A3 ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_1002_nonneg__linorder__cases,axiom,
! [A3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ~ ( ord_less_int @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_int @ B @ A3 ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_1003_not__less__real,axiom,
! [A3: real,B: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ~ ( ord_less_real @ B @ A3 ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ) ).
% not_less_real
thf(fact_1004_not__less__real,axiom,
! [A3: nat,B: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_nat @ B @ A3 ) )
= ( ord_less_eq_nat @ A3 @ B ) ) ) ) ).
% not_less_real
thf(fact_1005_not__less__real,axiom,
! [A3: int,B: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ~ ( ord_less_int @ B @ A3 ) )
= ( ord_less_eq_int @ A3 @ B ) ) ) ) ).
% not_less_real
thf(fact_1006_not__le__real,axiom,
! [A3: real,B: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ~ ( ord_less_eq_real @ B @ A3 ) )
= ( ord_less_real @ A3 @ B ) ) ) ) ).
% not_le_real
thf(fact_1007_not__le__real,axiom,
! [A3: nat,B: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ~ ( ord_less_eq_nat @ B @ A3 ) )
= ( ord_less_nat @ A3 @ B ) ) ) ) ).
% not_le_real
thf(fact_1008_not__le__real,axiom,
! [A3: int,B: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ~ ( ord_less_eq_int @ B @ A3 ) )
= ( ord_less_int @ A3 @ B ) ) ) ) ).
% not_le_real
thf(fact_1009_real__mult__less__cancel__right__pos,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_nat @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_1010_real__mult__less__cancel__right__pos,axiom,
! [A3: int,B: int,C: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ord_less_int @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_1011_real__mult__less__cancel__right__pos,axiom,
! [A3: real,B: real,C: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ord_less_real @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_1012_real__mult__less__cancel__left__pos,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_nat @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_1013_real__mult__less__cancel__left__pos,axiom,
! [A3: int,B: int,C: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_1014_real__mult__less__cancel__left__pos,axiom,
! [A3: real,B: real,C: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A3 @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_1015_real__mult__eq__0__iff,axiom,
! [A3: nat,B: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_1016_real__mult__eq__0__iff,axiom,
! [A3: int,B: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ( times_times_int @ A3 @ B )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_1017_real__mult__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_1018_semiring__real__line__class_Omult__neg__neg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_1019_semiring__real__line__class_Omult__neg__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_1020_semiring__real__line__class_Omult__neg__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_1021_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1022_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1023_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1024_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1025_mult__less__cancel__right__disj,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1026_mult__less__cancel__right__disj,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1027_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1028_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1029_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1030_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1031_mult__strict__right__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1032_mult__strict__right__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1033_mult__less__cancel__left__disj,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A3 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1034_mult__less__cancel__left__disj,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1035_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: a,B: a,C: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1036_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1037_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: int,B: int,C: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1038_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1039_mult__strict__left__mono__neg,axiom,
! [B: int,A3: int,C: int] :
( ( ord_less_int @ B @ A3 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1040_mult__strict__left__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_real @ B @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1041_mult__less__cancel__left__pos,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A3 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1042_mult__less__cancel__left__pos,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A3 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1043_mult__less__cancel__left__neg,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1044_mult__less__cancel__left__neg,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1045_zero__less__mult__pos2,axiom,
! [B: a,A3: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ B @ A3 ) )
=> ( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1046_zero__less__mult__pos2,axiom,
! [B: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1047_zero__less__mult__pos2,axiom,
! [B: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1048_zero__less__mult__pos2,axiom,
! [B: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1049_zero__less__mult__pos,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A3 @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1050_zero__less__mult__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1051_zero__less__mult__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1052_zero__less__mult__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1053_zero__less__mult__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1054_zero__less__mult__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1055_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ B @ A3 ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1056_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1057_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A3 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1058_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1059_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1060_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1061_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1062_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1063_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A3 @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1064_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1065_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1066_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1067_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: a,B: a] :
( ( ord_less_a @ A3 @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ ( times_times_a @ A3 @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1068_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1069_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1070_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1071_mult__less__0__iff,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1072_mult__less__0__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1073_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1074_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1075_linordered__ring__strict__class_Omult__neg__neg,axiom,
! [A3: int,B: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B ) ) ) ) ).
% linordered_ring_strict_class.mult_neg_neg
thf(fact_1076_linordered__ring__strict__class_Omult__neg__neg,axiom,
! [A3: real,B: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% linordered_ring_strict_class.mult_neg_neg
thf(fact_1077_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1078_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1079_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1080_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1081_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1082_vec__eq__iff,axiom,
( ( ^ [Y5: vec_a,Z: vec_a] : ( Y5 = Z ) )
= ( ^ [X4: vec_a,Y3: vec_a] :
( ( ( dim_vec_a @ X4 )
= ( dim_vec_a @ Y3 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_vec_a @ Y3 ) )
=> ( ( vec_index_a @ X4 @ I )
= ( vec_index_a @ Y3 @ I ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_1083_real__mult__le__cancel__right__pos,axiom,
! [A3: real,B: real,C: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_1084_real__mult__le__cancel__right__pos,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_eq_nat @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_1085_real__mult__le__cancel__right__pos,axiom,
! [A3: int,B: int,C: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ord_less_eq_int @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_1086_real__mult__le__cancel__left__pos,axiom,
! [A3: real,B: real,C: real] :
( ( ( ord_less_real @ A3 @ zero_zero_real )
| ( A3 = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A3 ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_1087_real__mult__le__cancel__left__pos,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ( ord_less_nat @ A3 @ zero_zero_nat )
| ( A3 = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A3 ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_eq_nat @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_1088_real__mult__le__cancel__left__pos,axiom,
! [A3: int,B: int,C: int] :
( ( ( ord_less_int @ A3 @ zero_zero_int )
| ( A3 = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A3 ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A3 @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_1089_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1090_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1091_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1092_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1093_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1094_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1095_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1096_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A3 @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1097_mult__right__le__imp__le,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1098_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1099_mult__right__le__imp__le,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1100_mult__right__le__imp__le,axiom,
! [A3: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A3 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1101_mult__left__le__imp__le,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1102_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1103_mult__left__le__imp__le,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1104_mult__left__le__imp__le,axiom,
! [C: a,A3: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A3 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1105_mult__le__cancel__left__pos,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A3 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1106_mult__le__cancel__left__pos,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A3 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1107_mult__le__cancel__left__neg,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1108_mult__le__cancel__left__neg,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1109_mult__less__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1110_mult__less__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1111_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1112_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1113_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1114_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_eq_a @ zero_zero_a @ A3 )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1115_mult__right__less__imp__less,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1116_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1117_mult__right__less__imp__less,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1118_mult__right__less__imp__less,axiom,
! [A3: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ C ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A3 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1119_mult__less__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1120_mult__less__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1121_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_real @ A3 @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1122_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1123_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: int,B: int,C: int,D: int] :
( ( ord_less_int @ A3 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1124_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: a,B: a,C: a,D: a] :
( ( ord_less_a @ A3 @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A3 @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1125_mult__left__less__imp__less,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1126_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1127_mult__left__less__imp__less,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1128_mult__left__less__imp__less,axiom,
! [C: a,A3: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A3 ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A3 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1129_mult__le__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1130_mult__le__cancel__right,axiom,
! [A3: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1131_mult__le__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1132_mult__le__cancel__left,axiom,
! [C: int,A3: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1133_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1134_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( A3 @ I3 ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( times_times_nat @ ( A3 @ I ) @ ( B @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_1135_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1136_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1137_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1138_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1139_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1140_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1141_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1142_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( F @ K3 @ I5 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1143_forall__finite_I1_J,axiom,
! [P: nat > $o,I5: nat] :
( ( ord_less_nat @ I5 @ zero_zero_nat )
=> ( P @ I5 ) ) ).
% forall_finite(1)
thf(fact_1144_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1145_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1146_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1147_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1148_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1149_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1150_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1151_int__ops_I7_J,axiom,
! [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1152_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1153_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq_int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq_int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_1154_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1155_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1156_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1157_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1158_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1159_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1160_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1161_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1162_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1163_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_1164_card__sum__le__nat__sum,axiom,
! [S: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_1165_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1166_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1167_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1168_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1169_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1170_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1171_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1172_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1173_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1174_pos__mult__pos__ge,axiom,
! [X: int,N: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1175_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X4 )
@ ^ [X4: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1176_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1177_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1178_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1179_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1180_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1181_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1182_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1183_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1184_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1185_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1186_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1187_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1188_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1189_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1190_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1191_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1192_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1193_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1194_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1195_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K6: nat] :
( N2
= ( plus_plus_nat @ M4 @ K6 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1196_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1197_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1198_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1199_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1200_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1201_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1202_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1203_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1204_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1205_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1206_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1207_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1208_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1209_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1210_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1211_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1212_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_1213_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1214_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1215_subset__card__intvl__is__intvl,axiom,
! [A: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) )
=> ( A
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1216_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1217_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1218_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M4: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M4 ) ) ) ) ).
% nat_less_real_le
thf(fact_1219_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1220_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A2: nat,B3: nat] :
( ( P @ A2 @ B3 )
= ( P @ B3 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B3: nat] :
( ( P @ A2 @ B3 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B3 ) ) )
=> ( P @ A3 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1221_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1222_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1223_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_1224_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1225_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1226_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1227_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1228_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1229_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1230_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1231_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_1232_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1233_diff__less__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1234_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1235_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1236_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1237_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1238_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1239_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1240_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1241_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1242_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1243_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1244_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1245_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1246_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1247_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1248_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1249_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1250_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1251_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1252_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1253_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1254_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1255_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1256_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1257_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1258_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1259_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1260_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1261_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B ) )
= ( ( ( ord_less_nat @ A3 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1262_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1263_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1264_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1265_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1266_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_a
@ ( groups1143116142660632562_nat_a
@ ^ [I: nat] : ( times_times_a @ ( vec_index_a @ y @ I ) @ ( vec_index_a @ ( mult_mat_vec_a @ a2 @ x ) @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_a @ ( mult_mat_vec_a @ a2 @ x ) ) ) )
@ ( groups1143116142660632562_nat_a
@ ^ [I: nat] : ( times_times_a @ ( vec_index_a @ y @ I ) @ ( vec_index_a @ b @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_vec_a @ b ) ) ) ) ).
%------------------------------------------------------------------------------