TPTP Problem File: SLH0590^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0038_Fishers_Inequality/prob_00200_010845__28206796_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1451 ( 714 unt; 177 typ; 0 def)
% Number of atoms : 3433 (1597 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12931 ( 285 ~; 85 |; 244 &;11119 @)
% ( 0 <=>;1198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 564 ( 564 >; 0 *; 0 +; 0 <<)
% Number of symbols : 157 ( 154 usr; 19 con; 0-3 aty)
% Number of variables : 3409 ( 136 ^;3208 !; 65 ?;3409 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:47.173
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Real__Oreal_J,type,
multiset_real: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Num__Onum_J,type,
multiset_num: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
multiset_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $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__List__Olist_Itf__a_J,type,
list_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__Num__Onum,type,
num: $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 (154)
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001t__Nat__Onat,type,
design2875492832550762736nt_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001tf__a,type,
design5397942185814921632port_a: multiset_set_a > set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Ointersection__numbers_001tf__a,type,
design3761797438660848528bers_a: multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001tf__a,type,
design1769254222028858111izes_a: multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_Ointersection__number_001tf__a,type,
design7842873109100088828mber_a: set_a > set_a > nat ).
thf(sy_c_Design__Basics_On__intersect__number_001tf__a,type,
design735257067508376852mber_a: set_a > nat > set_a > nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Nat__Onat,type,
design8239173135376323853nt_nat: set_nat > nat > set_nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001tf__a,type,
design2935547469388721088ocks_a: multiset_set_a > a > set_set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001tf__a,type,
design1146539425385464078lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001t__Nat__Onat,type,
design4269233978287968195nt_nat: set_nat > nat > set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
minus_3865385036109388885t_real: multiset_real > multiset_real > multiset_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
minus_minus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__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__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
plus_p8661369373666671365t_real: multiset_real > multiset_real > multiset_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
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__Num__Onum,type,
plus_plus_num: num > num > num ).
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__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
zero_z3170743180189231877et_int: multiset_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Num__Onum_J,type,
zero_z8056838136647266291et_num: multiset_num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
zero_z8811559133707751557t_real: multiset_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
zero_z3157962936165190495et_nat: multiset_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__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__Set__Oset_Itf__a_J_001t__Int__Oint,type,
groups6139252898804525648_a_int: ( set_a > int ) > set_set_a > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
groups6141743369313575924_a_nat: ( set_a > nat ) > set_set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
groups9174420418583655632a_real: ( set_a > real ) > set_set_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Int__Oint,type,
groups6332066207828071664_a_int: ( a > int ) > set_a > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Real__Oreal,type,
groups2740460157737275248a_real: ( a > real ) > set_a > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Multiset_Oadd__mset_001t__Int__Oint,type,
add_mset_int: int > multiset_int > multiset_int ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Num__Onum,type,
add_mset_num: num > multiset_num > multiset_num ).
thf(sy_c_Multiset_Oadd__mset_001t__Real__Oreal,type,
add_mset_real: real > multiset_real > multiset_real ).
thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_It__Nat__Onat_J,type,
add_mset_set_nat: set_nat > multiset_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_Itf__a_J,type,
add_mset_set_a: set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Real__Oreal,type,
mset_real: list_real > multiset_real ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Oset__mset_001t__Int__Oint,type,
set_mset_int: multiset_int > set_int ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Num__Onum,type,
set_mset_num: multiset_num > set_num ).
thf(sy_c_Multiset_Oset__mset_001t__Real__Oreal,type,
set_mset_real: multiset_real > set_real ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_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__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
size_s3818332516149715216t_real: multiset_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s6566526139600085008_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > 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__Multiset__Omultiset_It__Int__Oint_J,type,
ord_le1599922481286804176et_int: multiset_int > multiset_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Num__Onum_J,type,
ord_le6486017437744838590et_num: multiset_num > multiset_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
ord_le7573655249420395216t_real: multiset_real > multiset_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_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__Num__Onum,type,
ord_less_num: num > num > $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_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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__Multiset__Omultiset_It__Int__Oint_J,type,
ord_le2424384866860593884et_int: multiset_int > multiset_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Num__Onum_J,type,
ord_le7310479823318628298et_num: multiset_num > multiset_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Real__Oreal_J,type,
ord_le2426415917361421532t_real: multiset_real > multiset_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le4034546139768944438et_nat: multiset_set_nat > multiset_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_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__Num__Onum,type,
ord_less_eq_num: num > num > $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__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__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $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_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
set_or1222409239386451017an_num: num > num > set_num ).
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_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $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_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060B_062s,type,
b_s: list_set_a ).
thf(sy_v__092_060m_062,type,
m: nat ).
thf(sy_v_c,type,
c: nat > real ).
thf(sy_v_j_H,type,
j: nat ).
% Relevant facts (1270)
thf(fact_0_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ m ).
% assms(2)
thf(fact_1_cine0,axiom,
( ( c @ j )
!= zero_zero_real ) ).
% cine0
thf(fact_2_assms_I3_J,axiom,
ord_less_nat @ j @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% assms(3)
thf(fact_3_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_4_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_positive
thf(fact_5_innerge,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ ( power_power_real @ ( c @ J ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( ring_1_of_int_real @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ ( nth_set_a @ b_s @ J ) ) ) @ ( semiri1314217659103216013at_int @ m ) ) ) ) ) ) ).
% innerge
thf(fact_6_assms_I1_J,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% assms(1)
thf(fact_7_add__point__existing__blocks,axiom,
! [Bs: set_set_a,P: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ Bs )
=> ( member_a @ P @ Bl ) )
=> ( ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P @ Bs )
= ( mset_set_a @ b_s ) ) ) ).
% add_point_existing_blocks
thf(fact_8_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_9_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_10_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_11_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_12_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_13_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_14_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_15_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_16_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_17_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_18_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_19_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_20_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_21_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_22_sum__constant,axiom,
! [Y: nat,A2: set_a] :
( ( groups6334556678337121940_a_nat
@ ^ [X2: a] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_23_sum__constant,axiom,
! [Y: nat,A2: set_set_a] :
( ( groups6141743369313575924_a_nat
@ ^ [X2: set_a] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_set_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_24_sum__constant,axiom,
! [Y: int,A2: set_a] :
( ( groups6332066207828071664_a_int
@ ^ [X2: a] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_25_sum__constant,axiom,
! [Y: int,A2: set_set_a] :
( ( groups6139252898804525648_a_int
@ ^ [X2: set_a] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_set_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_26_sum__constant,axiom,
! [Y: int,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X2: nat] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_27_sum__constant,axiom,
! [Y: real,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [X2: a] : Y
@ A2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_28_sum__constant,axiom,
! [Y: real,A2: set_set_a] :
( ( groups9174420418583655632a_real
@ ^ [X2: set_a] : Y
@ A2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_a @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_29_sum__constant,axiom,
! [Y: real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : Y
@ A2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_30_sum__constant,axiom,
! [Y: nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_31_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_32_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_33_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_34_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_35_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_36_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_37_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_38_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_39_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_40_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_41_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_42_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_43_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_44_design__blocks__nempty,axiom,
( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a ) ).
% design_blocks_nempty
thf(fact_45_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_46_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_47_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_48_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_49_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_50_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_51_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_52_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_53_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_54_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_55_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_56_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_57_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_58_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_59_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_60_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_61_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_62_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_63_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_64_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_65_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_66_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_67_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_68_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_69_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_70_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_71_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_72_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_73_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_74_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_75_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_76_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_77_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_78_of__int__sum,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_79_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_80_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_81_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_82_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_83_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_84_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_85_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_86_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_87_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_88_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_96_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_97_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_98_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_99_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_100_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_101_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_102_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_103_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_104_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_105_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_106_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_107_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_108_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_109_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_110_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_111_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_112_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_113_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_int @ Z )
= ( numeral_numeral_int @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_114_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_real @ Z )
= ( numeral_numeral_real @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_115_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_116_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_real @ K ) ) ).
% of_int_numeral
thf(fact_117_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_118_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_119_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_120_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_121_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_122_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_123_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_124_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_125_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_126_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_127_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_128_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_129_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_130_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_131_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_132_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_133_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_134_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_135_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_136_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% of_int_power
thf(fact_137_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% of_int_power
thf(fact_138_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_139_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_140_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_141_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_142_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_143_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_144_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_145_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_146_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_147_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_148_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_149_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_150_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_151_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_152_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_153_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_154_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_155_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_156_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_157_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_158_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_159_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_160_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_161_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_162_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_163_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_164_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_165_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_166_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_167_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_168_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_169_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_170_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_171_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_172_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_173_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_174_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_175_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_176_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_177_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_178_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_179_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_180_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_181_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_182_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_183_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_184_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_185_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_186_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_187_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_188_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_189_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_190_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_191_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_192_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_193_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_194_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_195_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_196_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_197_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_198_assms_I4_J,axiom,
( zero_zero_real
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( power_power_real @ ( c @ J2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( ring_1_of_int_real @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ ( nth_set_a @ b_s @ J2 ) ) ) @ ( semiri1314217659103216013at_int @ m ) ) ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ m ) @ ( power_power_real @ ( groups6591440286371151544t_real @ c @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% assms(4)
thf(fact_199_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_200_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_201_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_202_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_203_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_204_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_205_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_206_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_207_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_208_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_209_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_210_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_211_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_212_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_213_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_214_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_215_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_216_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_217_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_218_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_219_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_220_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_221_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_222_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_223_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_224_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_225_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_226_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_227_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_228_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% int_sum
thf(fact_229_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_230_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_231_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_232_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_233_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_234_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_235_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_236_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_237_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_238_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_239_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_240_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_241_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_242_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_243_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_244_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_245_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_246_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_247_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_248_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_249_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_250_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_251_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_252_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_253_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_254_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_255_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_256_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_257_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_258_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_259_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_260_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_261_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_262_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_263_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_264_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_265_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_266_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_267_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_268_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_269_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_270_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_271_size__neq__size__imp__neq,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X )
!= ( size_s6566526139600085008_set_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_272_size__neq__size__imp__neq,axiom,
! [X: list_set_a,Y: list_set_a] :
( ( ( size_size_list_set_a @ X )
!= ( size_size_list_set_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_273_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_274_sum_Oreindex__bij__witness,axiom,
! [S2: set_a,I: nat > a,J: a > nat,T2: set_nat,H: nat > real,G: a > real] :
( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_a @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_275_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: nat > real,J: real > nat,T2: set_nat,H: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_real @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_276_sum_Oreindex__bij__witness,axiom,
! [S2: set_a,I: nat > a,J: a > nat,T2: set_nat,H: nat > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_a @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_277_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: nat > real,J: real > nat,T2: set_nat,H: nat > nat,G: real > nat] :
( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_real @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_278_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: a > nat,J: nat > a,T2: set_a,H: a > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_279_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: real > nat,J: nat > real,T2: set_real,H: real > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_real @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_280_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_281_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: a > nat,J: nat > a,T2: set_a,H: a > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_a @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups6334556678337121940_a_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_282_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: real > nat,J: nat > real,T2: set_real,H: real > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_real @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_283_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( ( J @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T2 )
=> ( member_nat @ ( I @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_284_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > a,A2: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_a @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_285_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_286_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > a,A2: set_a,H: a > nat,Gamma: nat > nat,Phi: a > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_a @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_287_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_288_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > nat,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_289_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_290_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_291_sum_Oeq__general__inverses,axiom,
! [B3: set_a,K: a > nat,A2: set_nat,H: nat > a,Gamma: a > nat,Phi: nat > nat] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups6334556678337121940_a_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_292_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_293_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_294_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_295_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_296_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_a,H: a > nat,Gamma: nat > nat,Phi: a > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_297_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_298_sum_Oeq__general,axiom,
! [B3: set_a,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_299_sum_Oeq__general,axiom,
! [B3: set_real,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_300_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_301_sum_Oeq__general,axiom,
! [B3: set_a,A2: set_nat,H: nat > a,Gamma: a > nat,Phi: nat > nat] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups6334556678337121940_a_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_302_sum_Oeq__general,axiom,
! [B3: set_real,A2: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_303_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_304_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > real,H: nat > real] :
( ( A2 = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_305_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A2 = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_306_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_307_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_308_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_309_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_310_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_311_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_312_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_313_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_314_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_315_sum_Oswap,axiom,
! [G: nat > nat > real,B3: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( G @ I4 ) @ B3 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ J2 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_316_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ J2 )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_317_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_318_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_319_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_320_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_321_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_322_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_323_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_324_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_325_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_326_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > nat,K3: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_327_sum__bounded__above__strict,axiom,
! [A2: set_a,F: a > nat,K3: nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( ord_less_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_328_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > int,K3: int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_329_sum__bounded__above__strict,axiom,
! [A2: set_a,F: a > int,K3: int] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( ord_less_int @ ( groups6332066207828071664_a_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_330_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > int,K3: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_331_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > real,K3: real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_332_sum__bounded__above__strict,axiom,
! [A2: set_a,F: a > real,K3: real] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( ord_less_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_333_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > real,K3: real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_334_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > nat,K3: nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_335_sum__bounded__above__strict,axiom,
! [A2: set_set_a,F: set_a > nat,K3: nat] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( ord_less_nat @ ( groups6141743369313575924_a_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_set_a @ A2 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_336_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_337_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_338_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_339_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > real,A2: set_a] :
( ( ( groups2740460157737275248a_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_340_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_341_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A2: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_342_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_343_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > int,A2: set_a] :
( ( ( groups6332066207828071664_a_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_344_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_345_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_346_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_347_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_348_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_a > real,A2: set_set_a] :
( ( ( groups9174420418583655632a_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_349_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_350_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_351_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_352_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_353_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_354_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_355_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_356_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_357_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_358_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_359_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_360_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_361_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_362_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_363_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_364_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_365_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_366_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_367_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_368_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_369_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_370_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_371_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_372_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_373_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_374_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_375_sum__mono,axiom,
! [K3: set_a,F: a > real,G: a > real] :
( ! [I2: a] :
( ( member_a @ I2 @ K3 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ K3 ) @ ( groups2740460157737275248a_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_376_sum__mono,axiom,
! [K3: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ K3 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K3 ) @ ( groups8097168146408367636l_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_377_sum__mono,axiom,
! [K3: set_a,F: a > nat,G: a > nat] :
( ! [I2: a] :
( ( member_a @ I2 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K3 ) @ ( groups6334556678337121940_a_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_378_sum__mono,axiom,
! [K3: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K3 ) @ ( groups1935376822645274424al_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_379_sum__mono,axiom,
! [K3: set_a,F: a > int,G: a > int] :
( ! [I2: a] :
( ( member_a @ I2 @ K3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ K3 ) @ ( groups6332066207828071664_a_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_380_sum__mono,axiom,
! [K3: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K3 ) @ ( groups1932886352136224148al_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_381_sum__mono,axiom,
! [K3: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K3 ) @ ( groups3539618377306564664at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_382_sum__mono,axiom,
! [K3: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K3 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K3 ) @ ( groups6591440286371151544t_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_383_sum__mono,axiom,
! [K3: set_nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_384_sum__mono,axiom,
! [K3: set_set_a,F: set_a > real,G: set_a > real] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ K3 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups9174420418583655632a_real @ F @ K3 ) @ ( groups9174420418583655632a_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_385_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J2 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_386_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J2 ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_387_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_388_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_389_sum__distrib__left,axiom,
! [R: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_390_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_391_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_392_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_393_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_394_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_395_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_396_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_397_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_398_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_399_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_400_sum__nonpos,axiom,
! [A2: set_a,F: a > real] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_401_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_402_sum__nonpos,axiom,
! [A2: set_a,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_403_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_404_sum__nonpos,axiom,
! [A2: set_a,F: a > int] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_405_sum__nonpos,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_406_sum__nonpos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_407_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_408_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_409_sum__nonpos,axiom,
! [A2: set_set_a,F: set_a > real] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups9174420418583655632a_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_410_sum__nonneg,axiom,
! [A2: set_a,F: a > real] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups2740460157737275248a_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_411_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_412_sum__nonneg,axiom,
! [A2: set_a,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_413_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_414_sum__nonneg,axiom,
! [A2: set_a,F: a > int] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups6332066207828071664_a_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_415_sum__nonneg,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_416_sum__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_417_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_418_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_419_sum__nonneg,axiom,
! [A2: set_set_a,F: set_a > real] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups9174420418583655632a_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_420_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_421_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_422_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_423_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_424_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_425_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_426_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_427_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_428_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_429_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_430_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_431_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_432_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_433_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_434_sum__bounded__below,axiom,
! [A2: set_real,K3: real,F: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_real @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K3 ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_435_sum__bounded__below,axiom,
! [A2: set_a,K3: real,F: a > real] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_real @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) @ K3 ) @ ( groups2740460157737275248a_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_436_sum__bounded__below,axiom,
! [A2: set_real,K3: nat,F: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_nat @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A2 ) ) @ K3 ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_437_sum__bounded__below,axiom,
! [A2: set_a,K3: nat,F: a > nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_nat @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_a @ A2 ) ) @ K3 ) @ ( groups6334556678337121940_a_nat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_438_sum__bounded__below,axiom,
! [A2: set_real,K3: int,F: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_int @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A2 ) ) @ K3 ) @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_439_sum__bounded__below,axiom,
! [A2: set_a,K3: int,F: a > int] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_int @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ A2 ) ) @ K3 ) @ ( groups6332066207828071664_a_int @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_440_sum__bounded__below,axiom,
! [A2: set_nat,K3: int,F: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_int @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K3 ) @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_441_sum__bounded__below,axiom,
! [A2: set_nat,K3: real,F: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_real @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) ) @ K3 ) @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_442_sum__bounded__below,axiom,
! [A2: set_nat,K3: nat,F: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_nat @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A2 ) ) @ K3 ) @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_443_sum__bounded__below,axiom,
! [A2: set_set_a,K3: real,F: set_a > real] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_real @ K3 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_a @ A2 ) ) @ K3 ) @ ( groups9174420418583655632a_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_444_sum__bounded__above,axiom,
! [A2: set_real,F: real > real,K3: real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_445_sum__bounded__above,axiom,
! [A2: set_a,F: a > real,K3: real] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_446_sum__bounded__above,axiom,
! [A2: set_real,F: real > nat,K3: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_447_sum__bounded__above,axiom,
! [A2: set_a,F: a > nat,K3: nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_448_sum__bounded__above,axiom,
! [A2: set_real,F: real > int,K3: int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_449_sum__bounded__above,axiom,
! [A2: set_a,F: a > int,K3: int] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_450_sum__bounded__above,axiom,
! [A2: set_nat,F: nat > int,K3: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_451_sum__bounded__above,axiom,
! [A2: set_nat,F: nat > real,K3: real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_452_sum__bounded__above,axiom,
! [A2: set_nat,F: nat > nat,K3: nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_453_sum__bounded__above,axiom,
! [A2: set_set_a,F: set_a > real,K3: real] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K3 ) )
=> ( ord_less_eq_real @ ( groups9174420418583655632a_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_a @ A2 ) ) @ K3 ) ) ) ).
% sum_bounded_above
thf(fact_454_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_455_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_456_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_457_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_458_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_459_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_460_power2__commute,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_461_power2__commute,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_462_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_463_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_464_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_465_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_466_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_467_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_468_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_469_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_470_simple__block__size__eq__card,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= ( finite_card_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ) ) ).
% simple_block_size_eq_card
thf(fact_471_const__intersect__block__size__diff,axiom,
! [J4: nat,J: nat] :
( ( ord_less_nat @ J4 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( J != J4 )
=> ( ( ( finite_card_a @ ( nth_set_a @ b_s @ J4 ) )
= m )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( finite_card_a @ ( nth_set_a @ b_s @ J ) ) @ m ) ) ) ) ) ) ) ).
% const_intersect_block_size_diff
thf(fact_472_Chebyshev__sum__upper,axiom,
! [N: nat,A: nat > int,B: nat > int] :
( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_int @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
=> ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_int @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_int
@ ( times_times_int @ ( semiri1314217659103216013at_int @ N )
@ ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( times_times_int @ ( A @ K4 ) @ ( B @ K4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_int @ ( groups3539618377306564664at_int @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3539618377306564664at_int @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_473_Chebyshev__sum__upper,axiom,
! [N: nat,A: nat > real,B: nat > real] :
( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_real @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
=> ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_real @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_real
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N )
@ ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( times_times_real @ ( A @ K4 ) @ ( B @ K4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_real @ ( groups6591440286371151544t_real @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups6591440286371151544t_real @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_474_of__int__hom_Ohom__power,axiom,
! [X: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ X @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ X ) @ N ) ) ).
% of_int_hom.hom_power
thf(fact_475_of__int__hom_Ohom__power,axiom,
! [X: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ X @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ X ) @ N ) ) ).
% of_int_hom.hom_power
thf(fact_476_inter__num__le__block__size,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_nat @ m @ ( finite_card_a @ Bl2 ) ) ) ) ).
% inter_num_le_block_size
thf(fact_477_of__int__hom_Ohom__minus,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_minus
thf(fact_478_of__int__hom_Ohom__minus,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_minus
thf(fact_479_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_480_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_481_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_482_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_483_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_484_atLeastLessThan__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_485_atLeastLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_486_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_487_atLeastLessThan__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_num @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_488_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_489_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_490_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_491_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_492_blocks__index__simp__unique,axiom,
! [I1: nat,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_size_list_set_a @ b_s ) )
=> ( ( ord_less_nat @ I22 @ ( size_size_list_set_a @ b_s ) )
=> ( ( I1 != I22 )
=> ( ( nth_set_a @ b_s @ I1 )
!= ( nth_set_a @ b_s @ I22 ) ) ) ) ) ).
% blocks_index_simp_unique
thf(fact_493_blocks__list__length,axiom,
( ( size_size_list_set_a @ b_s )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% blocks_list_length
thf(fact_494_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_495_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_496_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_497_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_498_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_499_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_500_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_501_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_502_Ring__Hom_Oof__int__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
= ( X = Y ) ) ).
% Ring_Hom.of_int_hom.eq_iff
thf(fact_503_block__set__nempty__imp__block__ex,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_504_design__support__def,axiom,
( ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) )
= ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ).
% design_support_def
thf(fact_505_dual__blocks__v,axiom,
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% dual_blocks_v
thf(fact_506_block__size__gt__0,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl2 ) ) ) ).
% block_size_gt_0
thf(fact_507_valid__blocks__index,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ ( nth_set_a @ b_s @ J ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% valid_blocks_index
thf(fact_508_valid__blocks__index__cons,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ? [J3: nat] :
( ( ( nth_set_a @ b_s @ J3 )
= Bl2 )
& ( ord_less_nat @ J3 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_cons
thf(fact_509_valid__blocks__index__obtains,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ! [J3: nat] :
~ ( ( ( nth_set_a @ b_s @ J3 )
= Bl2 )
& ( ord_less_nat @ J3 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_obtains
thf(fact_510_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_511_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_512_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_513_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_514_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_515_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_516_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_517_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_518_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_519_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_520_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_521_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_522_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_523_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_524_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_525_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_526_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_527_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_528_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_529_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_530_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_531_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_532_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_533_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_534_ivl__diff,axiom,
! [I: real,N: real,M: real] :
( ( ord_less_eq_real @ I @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M ) @ ( set_or66887138388493659n_real @ I @ N ) )
= ( set_or66887138388493659n_real @ N @ M ) ) ) ).
% ivl_diff
thf(fact_535_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_536_ivl__diff,axiom,
! [I: num,N: num,M: num] :
( ( ord_less_eq_num @ I @ N )
=> ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
= ( set_or1222409239386451017an_num @ N @ M ) ) ) ).
% ivl_diff
thf(fact_537_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_538_ivl__subset,axiom,
! [I: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_539_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_540_ivl__subset,axiom,
! [I: num,J: num,M: num,N: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
= ( ( ord_less_eq_num @ J @ I )
| ( ( ord_less_eq_num @ M @ I )
& ( ord_less_eq_num @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_541_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_542_of__int__hom_Ohom__add,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_543_of__int__hom_Ohom__add,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_544_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_545_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_546_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_547_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_548_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_549_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_550_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_551_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_552_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_553_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_554_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_555_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_556_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_557_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_558_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_559_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_560_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_561_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_562_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_563_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_564_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_565_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_566_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_567_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_hom.hom_zero
thf(fact_568_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_hom.hom_zero
thf(fact_569_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_570_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_571_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_572_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( times_times_int @ X @ Y ) )
= ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_573_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_574_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_575_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_576_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_577_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_578_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_579_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_580_of__int__hom_Ohom__sum,axiom,
! [F: nat > int,X5: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ X5 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
@ X5 ) ) ).
% of_int_hom.hom_sum
thf(fact_581_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_582_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_583_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_584_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_585_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_586_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_587_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_588_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_589_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_590_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_591_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_592_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_593_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_594_mult__hom_Ohom__add,axiom,
! [C: real,X: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_595_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_596_mult__hom_Ohom__add,axiom,
! [C: int,X: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_597_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_598_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_599_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_600_of__int__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_add_eq_zero
thf(fact_601_of__int__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_add_eq_zero
thf(fact_602_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_603_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_604_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_605_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_606_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_607_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_608_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_609_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_610_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_611_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_612_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_613_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y: real,C: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_614_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_615_mult__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int,C: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_616_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_617_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_618_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_619_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_620_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_621_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_622_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_623_sum__subtractf__nat,axiom,
! [A2: set_set_a,G: set_a > nat,F: set_a > nat] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups6141743369313575924_a_nat
@ ^ [X2: set_a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups6141743369313575924_a_nat @ F @ A2 ) @ ( groups6141743369313575924_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_624_sum__subtractf__nat,axiom,
! [A2: set_a,G: a > nat,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_625_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_626_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_627_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_628_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_629_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_630_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_631_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_632_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_633_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_634_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_635_sum_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P ) ) )
= ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_636_sum_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ P ) ) )
= ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_637_sum_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P ) ) )
= ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_638_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_639_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_640_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_641_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_642_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_643_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_644_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_645_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_646_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_647_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_648_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_649_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_650_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_651_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_652_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_653_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_654_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_655_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_656_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M5: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_657_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z3: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_658_of__int__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
=> ( X = Y ) ) ).
% of_int_hom.injectivity
thf(fact_659_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_660_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_661_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_662_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_663_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_664_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_665_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_666_sum__power2__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_667_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_668_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_669_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_670_power2__sum,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_671_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_672_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_673_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_674_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_675_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_676_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_677_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_678_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_679_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_680_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_681_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_682_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_683_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_684_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_685_atLeastLessThan__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_eq_num @ B @ A )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_686_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_687_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_688_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_689_atLeastLessThan__inj_I2_J,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_690_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_691_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_692_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_693_atLeastLessThan__inj_I1_J,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_694_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_695_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_696_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_697_Ico__eq__Ico,axiom,
! [L: num,H: num,L2: num,H2: num] :
( ( ( set_or1222409239386451017an_num @ L @ H )
= ( set_or1222409239386451017an_num @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_num @ L @ H )
& ~ ( ord_less_num @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_698_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_699_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_700_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_701_atLeastLessThan__eq__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_702_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_703_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_704_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_705_power2__diff,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_706_power2__diff,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_707_mult__hom_Ohom__sum,axiom,
! [C: real,F: nat > real,X5: set_nat] :
( ( times_times_real @ C @ ( groups6591440286371151544t_real @ F @ X5 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( times_times_real @ C @ ( F @ X2 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_708_mult__hom_Ohom__sum,axiom,
! [C: nat,F: nat > nat,X5: set_nat] :
( ( times_times_nat @ C @ ( groups3542108847815614940at_nat @ F @ X5 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( times_times_nat @ C @ ( F @ X2 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_709_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_710_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_711_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_712_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_713_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_714_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_715_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_716_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_717_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_718_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_719_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_720_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_721_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_722_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_723_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_724_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_725_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_726_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_727_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_728_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_729_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_730_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_731_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_732_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > real,H: nat > real] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( ( ord_less_nat @ X3 @ D )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups6591440286371151544t_real @ H @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_733_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H: nat > nat] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( ( ord_less_nat @ X3 @ D )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups3542108847815614940at_nat @ H @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_734_card__2__iff_H,axiom,
! [S2: set_a] :
( ( ( finite_card_a @ S2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ S2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ S2 )
& ( X2 != Y4 )
& ! [Z4: a] :
( ( member_a @ Z4 @ S2 )
=> ( ( Z4 = X2 )
| ( Z4 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_735_card__2__iff_H,axiom,
! [S2: set_set_a] :
( ( ( finite_card_set_a @ S2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ S2 )
& ? [Y4: set_a] :
( ( member_set_a @ Y4 @ S2 )
& ( X2 != Y4 )
& ! [Z4: set_a] :
( ( member_set_a @ Z4 @ S2 )
=> ( ( Z4 = X2 )
| ( Z4 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_736_card__2__iff_H,axiom,
! [S2: set_nat] :
( ( ( finite_card_nat @ S2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ S2 )
& ( X2 != Y4 )
& ! [Z4: nat] :
( ( member_nat @ Z4 @ S2 )
=> ( ( Z4 = X2 )
| ( Z4 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_737_sum__diff__nat__ivl,axiom,
! [M: nat,N: nat,P: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P )
=> ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N @ P ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_738_sum__diff__nat__ivl,axiom,
! [M: nat,N: nat,P: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P )
=> ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ P ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ N @ P ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_739_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
=> ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_740_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 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_741_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_742_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_743_block__size__inter__num__cases,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ m @ ( finite_card_a @ Bl2 ) )
| ( ( ( finite_card_a @ Bl2 )
= m )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ Bl2 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ord_less_nat @ m @ ( finite_card_a @ X4 ) ) ) ) ) ) ) ).
% block_size_inter_num_cases
thf(fact_744_max__one__block__size__inter,axiom,
! [Bl2: set_a,Bl22: set_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( finite_card_a @ Bl2 )
= m )
=> ( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ Bl2 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ord_less_nat @ m @ ( finite_card_a @ Bl22 ) ) ) ) ) ) ).
% max_one_block_size_inter
thf(fact_745_block__sizes__non__empty,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% block_sizes_non_empty
thf(fact_746_obtains__two__diff__block__indexes,axiom,
! [J1: nat,J22: nat] :
( ( ord_less_nat @ J1 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ J22 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( J1 != J22 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ! [Bl1: set_a] :
( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( nth_set_a @ b_s @ J1 )
= Bl1 )
=> ! [Bl23: set_a] :
( ( member_set_a @ Bl23 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ Bl1 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( nth_set_a @ b_s @ J22 )
!= Bl23 ) ) ) ) ) ) ) ) ).
% obtains_two_diff_block_indexes
thf(fact_747_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_748_sum__squares__bound,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_749_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_750_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_751_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X @ Y ) )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_752_empty__eq__union,axiom,
! [M5: multiset_set_a,N4: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M5 @ N4 ) )
= ( ( M5 = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_753_union__eq__empty,axiom,
! [M5: multiset_set_a,N4: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M5 @ N4 )
= zero_z5079479921072680283_set_a )
= ( ( M5 = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_754_union__mset__add__mset__right,axiom,
! [A2: multiset_set_a,A: set_a,B3: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A2 @ ( add_mset_set_a @ A @ B3 ) )
= ( add_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A2 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_755_union__mset__add__mset__left,axiom,
! [A: set_a,A2: multiset_set_a,B3: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ A @ A2 ) @ B3 )
= ( add_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A2 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_756_add__mset__add__mset__same__iff,axiom,
! [A: set_a,A2: multiset_set_a,B3: multiset_set_a] :
( ( ( add_mset_set_a @ A @ A2 )
= ( add_mset_set_a @ A @ B3 ) )
= ( A2 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_757_multi__self__add__other__not__self,axiom,
! [M5: multiset_set_a,X: set_a] :
( M5
!= ( add_mset_set_a @ X @ M5 ) ) ).
% multi_self_add_other_not_self
thf(fact_758_diff__diff__add__mset,axiom,
! [M5: multiset_set_a,N4: multiset_set_a,P2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M5 @ N4 ) @ P2 )
= ( minus_706656509937749387_set_a @ M5 @ ( plus_p2331992037799027419_set_a @ N4 @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_759_sys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ( finite_card_a @ X3 )
= X ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_760_sys__block__sizes__in,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_nat @ ( finite_card_a @ Bl2 ) @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ).
% sys_block_sizes_in
thf(fact_761_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_762_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_763_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_764_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_765_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_766_size__union,axiom,
! [M5: multiset_set_a,N4: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N4 ) )
= ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M5 ) @ ( size_s6566526139600085008_set_a @ N4 ) ) ) ).
% size_union
thf(fact_767_single__eq__single,axiom,
! [A: set_a,B: set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_768_add__mset__eq__single,axiom,
! [B: set_a,M5: multiset_set_a,A: set_a] :
( ( ( add_mset_set_a @ B @ M5 )
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( ( B = A )
& ( M5 = zero_z5079479921072680283_set_a ) ) ) ).
% add_mset_eq_single
thf(fact_769_single__eq__add__mset,axiom,
! [A: set_a,B: set_a,M5: multiset_set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( add_mset_set_a @ B @ M5 ) )
= ( ( B = A )
& ( M5 = zero_z5079479921072680283_set_a ) ) ) ).
% single_eq_add_mset
thf(fact_770_add__mset__eq__singleton__iff,axiom,
! [X: set_a,M5: multiset_set_a,Y: set_a] :
( ( ( add_mset_set_a @ X @ M5 )
= ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
= ( ( M5 = zero_z5079479921072680283_set_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_771_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_772_blocks__index__ne__belong,axiom,
! [I1: nat,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_size_list_set_a @ b_s ) )
=> ( ( ord_less_nat @ I22 @ ( size_size_list_set_a @ b_s ) )
=> ( ( I1 != I22 )
=> ( member_set_a @ ( nth_set_a @ b_s @ I22 ) @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ ( nth_set_a @ b_s @ I1 ) @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% blocks_index_ne_belong
thf(fact_773_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_774_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_775_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_776_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_777_size__eq__0__iff__empty,axiom,
! [M5: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M5 )
= zero_zero_nat )
= ( M5 = zero_z5079479921072680283_set_a ) ) ).
% size_eq_0_iff_empty
thf(fact_778_size__mset,axiom,
! [Xs: list_set_a] :
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% size_mset
thf(fact_779_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_780_diff__add__mset__swap,axiom,
! [B: a,A2: multiset_a,M5: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M5 ) @ A2 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M5 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_781_diff__add__mset__swap,axiom,
! [B: real,A2: multiset_real,M5: multiset_real] :
( ~ ( member_real @ B @ ( set_mset_real @ A2 ) )
=> ( ( minus_3865385036109388885t_real @ ( add_mset_real @ B @ M5 ) @ A2 )
= ( add_mset_real @ B @ ( minus_3865385036109388885t_real @ M5 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_782_diff__add__mset__swap,axiom,
! [B: nat,A2: multiset_nat,M5: multiset_nat] :
( ~ ( member_nat @ B @ ( set_mset_nat @ A2 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ M5 ) @ A2 )
= ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ M5 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_783_diff__add__mset__swap,axiom,
! [B: set_a,A2: multiset_set_a,M5: multiset_set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M5 ) @ A2 )
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M5 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_784_add__mset__remove__trivial,axiom,
! [X: set_a,M5: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M5 ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M5 ) ).
% add_mset_remove_trivial
thf(fact_785_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_786_diff__single__trivial,axiom,
! [X: a,M5: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M5 ) ) ).
% diff_single_trivial
thf(fact_787_diff__single__trivial,axiom,
! [X: real,M5: multiset_real] :
( ~ ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) )
= M5 ) ) ).
% diff_single_trivial
thf(fact_788_diff__single__trivial,axiom,
! [X: nat,M5: multiset_nat] :
( ~ ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= M5 ) ) ).
% diff_single_trivial
thf(fact_789_diff__single__trivial,axiom,
! [X: set_a,M5: multiset_set_a] :
( ~ ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M5 ) ) ).
% diff_single_trivial
thf(fact_790_diff__union__swap2,axiom,
! [Y: a,M5: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M5 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M5 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_791_diff__union__swap2,axiom,
! [Y: real,M5: multiset_real,X: real] :
( ( member_real @ Y @ ( set_mset_real @ M5 ) )
=> ( ( minus_3865385036109388885t_real @ ( add_mset_real @ X @ M5 ) @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
= ( add_mset_real @ X @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) ) ) ) ) ).
% diff_union_swap2
thf(fact_792_diff__union__swap2,axiom,
! [Y: nat,M5: multiset_nat,X: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ M5 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X @ M5 ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_swap2
thf(fact_793_diff__union__swap2,axiom,
! [Y: set_a,M5: multiset_set_a,X: set_a] :
( ( member_set_a @ Y @ ( set_mset_set_a @ M5 ) )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M5 ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
= ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_794_insert__DiffM,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M5 ) ) ).
% insert_DiffM
thf(fact_795_insert__DiffM,axiom,
! [X: real,M5: multiset_real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ( add_mset_real @ X @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) )
= M5 ) ) ).
% insert_DiffM
thf(fact_796_insert__DiffM,axiom,
! [X: nat,M5: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= M5 ) ) ).
% insert_DiffM
thf(fact_797_insert__DiffM,axiom,
! [X: set_a,M5: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= M5 ) ) ).
% insert_DiffM
thf(fact_798_power__add__numeral2,axiom,
! [A: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_799_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_800_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_801_power__add__numeral,axiom,
! [A: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_802_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_803_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_804_empty__neutral_I2_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X @ zero_z5079479921072680283_set_a )
= X ) ).
% empty_neutral(2)
thf(fact_805_empty__neutral_I1_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_806_insert__DiffM2,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M5 ) ) ).
% insert_DiffM2
thf(fact_807_insert__DiffM2,axiom,
! [X: real,M5: multiset_real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ( plus_p8661369373666671365t_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) )
= M5 ) ) ).
% insert_DiffM2
thf(fact_808_insert__DiffM2,axiom,
! [X: nat,M5: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= M5 ) ) ).
% insert_DiffM2
thf(fact_809_insert__DiffM2,axiom,
! [X: set_a,M5: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M5 ) ) ).
% insert_DiffM2
thf(fact_810_multiset__cases,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ~ ! [X3: set_a,N5: multiset_set_a] :
( M5
!= ( add_mset_set_a @ X3 @ N5 ) ) ) ).
% multiset_cases
thf(fact_811_diff__union__swap,axiom,
! [A: set_a,B: set_a,M5: multiset_set_a] :
( ( A != B )
=> ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M5 ) @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ).
% diff_union_swap
thf(fact_812_multiset__induct,axiom,
! [P2: multiset_set_a > $o,M5: multiset_set_a] :
( ( P2 @ zero_z5079479921072680283_set_a )
=> ( ! [X3: set_a,M6: multiset_set_a] :
( ( P2 @ M6 )
=> ( P2 @ ( add_mset_set_a @ X3 @ M6 ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct
thf(fact_813_single__is__union,axiom,
! [A: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( plus_p2331992037799027419_set_a @ M5 @ N4 ) )
= ( ( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= M5 )
& ( N4 = zero_z5079479921072680283_set_a ) )
| ( ( M5 = zero_z5079479921072680283_set_a )
& ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= N4 ) ) ) ) ).
% single_is_union
thf(fact_814_union__is__single,axiom,
! [M5: multiset_set_a,N4: multiset_set_a,A: set_a] :
( ( ( plus_p2331992037799027419_set_a @ M5 @ N4 )
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( ( ( M5
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
& ( N4 = zero_z5079479921072680283_set_a ) )
| ( ( M5 = zero_z5079479921072680283_set_a )
& ( N4
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% union_is_single
thf(fact_815_add__eq__conv__diff,axiom,
! [A: set_a,M5: multiset_set_a,B: set_a,N4: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M5 )
= ( add_mset_set_a @ B @ N4 ) )
= ( ( ( M5 = N4 )
& ( A = B ) )
| ( ( M5
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
& ( N4
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_816_multiset__induct2,axiom,
! [P2: multiset_set_a > multiset_set_a > $o,M5: multiset_set_a,N4: multiset_set_a] :
( ( P2 @ zero_z5079479921072680283_set_a @ zero_z5079479921072680283_set_a )
=> ( ! [A3: set_a,M6: multiset_set_a,N5: multiset_set_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ ( add_mset_set_a @ A3 @ M6 ) @ N5 ) )
=> ( ! [A3: set_a,M6: multiset_set_a,N5: multiset_set_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ M6 @ ( add_mset_set_a @ A3 @ N5 ) ) )
=> ( P2 @ M5 @ N4 ) ) ) ) ).
% multiset_induct2
thf(fact_817_multi__member__skip,axiom,
! [X: a,XS: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ XS ) )
=> ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_818_multi__member__skip,axiom,
! [X: real,XS: multiset_real,Y: real] :
( ( member_real @ X @ ( set_mset_real @ XS ) )
=> ( member_real @ X @ ( set_mset_real @ ( plus_p8661369373666671365t_real @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_819_multi__member__skip,axiom,
! [X: nat,XS: multiset_nat,Y: nat] :
( ( member_nat @ X @ ( set_mset_nat @ XS ) )
=> ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_820_multi__member__skip,axiom,
! [X: set_a,XS: multiset_set_a,Y: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ XS ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_821_multi__member__this,axiom,
! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_822_multi__member__this,axiom,
! [X: real,XS: multiset_real] : ( member_real @ X @ ( set_mset_real @ ( plus_p8661369373666671365t_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) @ XS ) ) ) ).
% multi_member_this
thf(fact_823_multi__member__this,axiom,
! [X: nat,XS: multiset_nat] : ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).
% multi_member_this
thf(fact_824_multi__member__this,axiom,
! [X: set_a,XS: multiset_set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_825_empty__not__add__mset,axiom,
! [A: set_a,A2: multiset_set_a] :
( zero_z5079479921072680283_set_a
!= ( add_mset_set_a @ A @ A2 ) ) ).
% empty_not_add_mset
thf(fact_826_add__mset__add__single,axiom,
( add_mset_set_a
= ( ^ [A4: set_a,A5: multiset_set_a] : ( plus_p2331992037799027419_set_a @ A5 @ ( add_mset_set_a @ A4 @ zero_z5079479921072680283_set_a ) ) ) ) ).
% add_mset_add_single
thf(fact_827_multi__nonempty__split,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ? [A6: multiset_set_a,A3: set_a] :
( M5
= ( add_mset_set_a @ A3 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_828_union__single__eq__diff,axiom,
! [X: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( ( add_mset_set_a @ X @ M5 )
= N4 )
=> ( M5
= ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_829_diff__union__single__conv,axiom,
! [A: a,J5: multiset_a,I5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ J5 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I5 @ J5 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I5 @ ( minus_3765977307040488491iset_a @ J5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_830_diff__union__single__conv,axiom,
! [A: real,J5: multiset_real,I5: multiset_real] :
( ( member_real @ A @ ( set_mset_real @ J5 ) )
=> ( ( minus_3865385036109388885t_real @ ( plus_p8661369373666671365t_real @ I5 @ J5 ) @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) )
= ( plus_p8661369373666671365t_real @ I5 @ ( minus_3865385036109388885t_real @ J5 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_831_diff__union__single__conv,axiom,
! [A: nat,J5: multiset_nat,I5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ J5 ) )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I5 @ J5 ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ I5 @ ( minus_8522176038001411705et_nat @ J5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_832_diff__union__single__conv,axiom,
! [A: set_a,J5: multiset_set_a,I5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ J5 ) )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ I5 @ J5 ) @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( plus_p2331992037799027419_set_a @ I5 @ ( minus_706656509937749387_set_a @ J5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_833_add__mset__diff__bothsides,axiom,
! [A: set_a,M5: multiset_set_a,A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ A @ M5 ) @ ( add_mset_set_a @ A @ A2 ) )
= ( minus_706656509937749387_set_a @ M5 @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_834_union__single__eq__member,axiom,
! [X: a,M5: multiset_a,N4: multiset_a] :
( ( ( add_mset_a @ X @ M5 )
= N4 )
=> ( member_a @ X @ ( set_mset_a @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_835_union__single__eq__member,axiom,
! [X: real,M5: multiset_real,N4: multiset_real] :
( ( ( add_mset_real @ X @ M5 )
= N4 )
=> ( member_real @ X @ ( set_mset_real @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_836_union__single__eq__member,axiom,
! [X: nat,M5: multiset_nat,N4: multiset_nat] :
( ( ( add_mset_nat @ X @ M5 )
= N4 )
=> ( member_nat @ X @ ( set_mset_nat @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_837_union__single__eq__member,axiom,
! [X: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( ( add_mset_set_a @ X @ M5 )
= N4 )
=> ( member_set_a @ X @ ( set_mset_set_a @ N4 ) ) ) ).
% union_single_eq_member
thf(fact_838_insert__noteq__member,axiom,
! [B: a,B3: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B @ B3 )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_839_insert__noteq__member,axiom,
! [B: real,B3: multiset_real,C: real,C2: multiset_real] :
( ( ( add_mset_real @ B @ B3 )
= ( add_mset_real @ C @ C2 ) )
=> ( ( B != C )
=> ( member_real @ C @ ( set_mset_real @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_840_insert__noteq__member,axiom,
! [B: nat,B3: multiset_nat,C: nat,C2: multiset_nat] :
( ( ( add_mset_nat @ B @ B3 )
= ( add_mset_nat @ C @ C2 ) )
=> ( ( B != C )
=> ( member_nat @ C @ ( set_mset_nat @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_841_insert__noteq__member,axiom,
! [B: set_a,B3: multiset_set_a,C: set_a,C2: multiset_set_a] :
( ( ( add_mset_set_a @ B @ B3 )
= ( add_mset_set_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_set_a @ C @ ( set_mset_set_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_842_multi__member__split,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ? [A6: multiset_a] :
( M5
= ( add_mset_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_843_multi__member__split,axiom,
! [X: real,M5: multiset_real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ? [A6: multiset_real] :
( M5
= ( add_mset_real @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_844_multi__member__split,axiom,
! [X: nat,M5: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ? [A6: multiset_nat] :
( M5
= ( add_mset_nat @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_845_multi__member__split,axiom,
! [X: set_a,M5: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ? [A6: multiset_set_a] :
( M5
= ( add_mset_set_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_846_diff__union__cancelR,axiom,
! [M5: multiset_set_a,N4: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N4 ) @ N4 )
= M5 ) ).
% diff_union_cancelR
thf(fact_847_diff__union__cancelL,axiom,
! [N4: multiset_set_a,M5: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ N4 @ M5 ) @ N4 )
= M5 ) ).
% diff_union_cancelL
thf(fact_848_Multiset_Odiff__right__commute,axiom,
! [M5: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M5 @ N4 ) @ Q )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M5 @ Q ) @ N4 ) ) ).
% Multiset.diff_right_commute
thf(fact_849_union__iff,axiom,
! [A: a,A2: multiset_a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B3 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A2 ) )
| ( member_a @ A @ ( set_mset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_850_union__iff,axiom,
! [A: real,A2: multiset_real,B3: multiset_real] :
( ( member_real @ A @ ( set_mset_real @ ( plus_p8661369373666671365t_real @ A2 @ B3 ) ) )
= ( ( member_real @ A @ ( set_mset_real @ A2 ) )
| ( member_real @ A @ ( set_mset_real @ B3 ) ) ) ) ).
% union_iff
thf(fact_851_union__iff,axiom,
! [A: nat,A2: multiset_nat,B3: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B3 ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
| ( member_nat @ A @ ( set_mset_nat @ B3 ) ) ) ) ).
% union_iff
thf(fact_852_union__iff,axiom,
! [A: set_a,A2: multiset_set_a,B3: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B3 ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
| ( member_set_a @ A @ ( set_mset_set_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_853_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B4: multiset_a] :
( A2
!= ( add_mset_a @ A @ B4 ) ) ) ).
% mset_add
thf(fact_854_mset__add,axiom,
! [A: real,A2: multiset_real] :
( ( member_real @ A @ ( set_mset_real @ A2 ) )
=> ~ ! [B4: multiset_real] :
( A2
!= ( add_mset_real @ A @ B4 ) ) ) ).
% mset_add
thf(fact_855_mset__add,axiom,
! [A: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ~ ! [B4: multiset_nat] :
( A2
!= ( add_mset_nat @ A @ B4 ) ) ) ).
% mset_add
thf(fact_856_mset__add,axiom,
! [A: set_a,A2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
=> ~ ! [B4: multiset_set_a] :
( A2
!= ( add_mset_set_a @ A @ B4 ) ) ) ).
% mset_add
thf(fact_857_Multiset_Odiff__add,axiom,
! [M5: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ M5 @ ( plus_p2331992037799027419_set_a @ N4 @ Q ) )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M5 @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_858_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_859_add__eq__conv__ex,axiom,
! [A: set_a,M5: multiset_set_a,B: set_a,N4: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M5 )
= ( add_mset_set_a @ B @ N4 ) )
= ( ( ( M5 = N4 )
& ( A = B ) )
| ? [K5: multiset_set_a] :
( ( M5
= ( add_mset_set_a @ B @ K5 ) )
& ( N4
= ( add_mset_set_a @ A @ K5 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_860_add__mset__commute,axiom,
! [X: set_a,Y: set_a,M5: multiset_set_a] :
( ( add_mset_set_a @ X @ ( add_mset_set_a @ Y @ M5 ) )
= ( add_mset_set_a @ Y @ ( add_mset_set_a @ X @ M5 ) ) ) ).
% add_mset_commute
thf(fact_861_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_862_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_863_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_864_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_865_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_866_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_867_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_868_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_869_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_870_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_871_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_872_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_873_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_874_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_875_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_876_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_877_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_878_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_879_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_880_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_881_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_882_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_883_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_884_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_885_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_886_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_887_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_888_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_889_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_890_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_891_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_892_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_893_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_894_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_895_more__than__one__mset__mset__diff,axiom,
! [A: a,M5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M5 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_896_more__than__one__mset__mset__diff,axiom,
! [A: real,M5: multiset_real] :
( ( member_real @ A @ ( set_mset_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) ) )
=> ( ( set_mset_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) )
= ( set_mset_real @ M5 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_897_more__than__one__mset__mset__diff,axiom,
! [A: nat,M5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( set_mset_nat @ M5 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_898_more__than__one__mset__mset__diff,axiom,
! [A: set_a,M5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( set_mset_set_a @ M5 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_899_multiset__add__sub__el__shuffle,axiom,
! [C: a,B3: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B3 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B3 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_900_multiset__add__sub__el__shuffle,axiom,
! [C: real,B3: multiset_real,B: real] :
( ( member_real @ C @ ( set_mset_real @ B3 ) )
=> ( ( B != C )
=> ( ( add_mset_real @ B @ ( minus_3865385036109388885t_real @ B3 @ ( add_mset_real @ C @ zero_z8811559133707751557t_real ) ) )
= ( minus_3865385036109388885t_real @ ( add_mset_real @ B @ B3 ) @ ( add_mset_real @ C @ zero_z8811559133707751557t_real ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_901_multiset__add__sub__el__shuffle,axiom,
! [C: nat,B3: multiset_nat,B: nat] :
( ( member_nat @ C @ ( set_mset_nat @ B3 ) )
=> ( ( B != C )
=> ( ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ B3 ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_902_multiset__add__sub__el__shuffle,axiom,
! [C: set_a,B3: multiset_set_a,B: set_a] :
( ( member_set_a @ C @ ( set_mset_set_a @ B3 ) )
=> ( ( B != C )
=> ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) )
= ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ B3 ) @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_903_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_a,A: a] :
( ( N4
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_904_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_real,A: real] :
( ( N4
= ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ N4 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) ) )
= ( member_real @ A @ ( set_mset_real @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_905_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_nat,A: nat] :
( ( N4
= ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A @ ( set_mset_nat @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_906_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_set_a,A: set_a] :
( ( N4
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
= ( member_set_a @ A @ ( set_mset_set_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_907_add__mset__remove__trivial__If,axiom,
! [A: a,N4: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N4 ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_908_add__mset__remove__trivial__If,axiom,
! [A: real,N4: multiset_real] :
( ( ( member_real @ A @ ( set_mset_real @ N4 ) )
=> ( ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ N4 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) )
= N4 ) )
& ( ~ ( member_real @ A @ ( set_mset_real @ N4 ) )
=> ( ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ N4 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) )
= ( add_mset_real @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_909_add__mset__remove__trivial__If,axiom,
! [A: nat,N4: multiset_nat] :
( ( ( member_nat @ A @ ( set_mset_nat @ N4 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= N4 ) )
& ( ~ ( member_nat @ A @ ( set_mset_nat @ N4 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( add_mset_nat @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_910_add__mset__remove__trivial__If,axiom,
! [A: set_a,N4: multiset_set_a] :
( ( ( member_set_a @ A @ ( set_mset_set_a @ N4 ) )
=> ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= N4 ) )
& ( ~ ( member_set_a @ A @ ( set_mset_set_a @ N4 ) )
=> ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( add_mset_set_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_911_multi__drop__mem__not__eq,axiom,
! [C: a,B3: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B3 ) )
=> ( ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_912_multi__drop__mem__not__eq,axiom,
! [C: real,B3: multiset_real] :
( ( member_real @ C @ ( set_mset_real @ B3 ) )
=> ( ( minus_3865385036109388885t_real @ B3 @ ( add_mset_real @ C @ zero_z8811559133707751557t_real ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_913_multi__drop__mem__not__eq,axiom,
! [C: nat,B3: multiset_nat] :
( ( member_nat @ C @ ( set_mset_nat @ B3 ) )
=> ( ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_914_multi__drop__mem__not__eq,axiom,
! [C: set_a,B3: multiset_set_a] :
( ( member_set_a @ C @ ( set_mset_set_a @ B3 ) )
=> ( ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_915_diff__single__eq__union,axiom,
! [X: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N4 )
= ( M5
= ( add_mset_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_916_diff__single__eq__union,axiom,
! [X: real,M5: multiset_real,N4: multiset_real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ( ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) )
= N4 )
= ( M5
= ( add_mset_real @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_917_diff__single__eq__union,axiom,
! [X: nat,M5: multiset_nat,N4: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ( ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= N4 )
= ( M5
= ( add_mset_nat @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_918_diff__single__eq__union,axiom,
! [X: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ( ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= N4 )
= ( M5
= ( add_mset_set_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_919_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_920_multi__member__last,axiom,
! [X: real] : ( member_real @ X @ ( set_mset_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) ) ).
% multi_member_last
thf(fact_921_multi__member__last,axiom,
! [X: nat] : ( member_nat @ X @ ( set_mset_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_922_multi__member__last,axiom,
! [X: set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ).
% multi_member_last
thf(fact_923_multiset__induct__min,axiom,
! [P2: multiset_real > $o,M5: multiset_real] :
( ( P2 @ zero_z8811559133707751557t_real )
=> ( ! [X3: real,M6: multiset_real] :
( ( P2 @ M6 )
=> ( ! [Xa: real] :
( ( member_real @ Xa @ ( set_mset_real @ M6 ) )
=> ( ord_less_eq_real @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_real @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_924_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M5: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P2 @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_925_multiset__induct__min,axiom,
! [P2: multiset_int > $o,M5: multiset_int] :
( ( P2 @ zero_z3170743180189231877et_int )
=> ( ! [X3: int,M6: multiset_int] :
( ( P2 @ M6 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M6 ) )
=> ( ord_less_eq_int @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_int @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_926_multiset__induct__min,axiom,
! [P2: multiset_num > $o,M5: multiset_num] :
( ( P2 @ zero_z8056838136647266291et_num )
=> ( ! [X3: num,M6: multiset_num] :
( ( P2 @ M6 )
=> ( ! [Xa: num] :
( ( member_num @ Xa @ ( set_mset_num @ M6 ) )
=> ( ord_less_eq_num @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_num @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_min
thf(fact_927_multiset__induct__max,axiom,
! [P2: multiset_real > $o,M5: multiset_real] :
( ( P2 @ zero_z8811559133707751557t_real )
=> ( ! [X3: real,M6: multiset_real] :
( ( P2 @ M6 )
=> ( ! [Xa: real] :
( ( member_real @ Xa @ ( set_mset_real @ M6 ) )
=> ( ord_less_eq_real @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_real @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_928_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M5: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P2 @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_929_multiset__induct__max,axiom,
! [P2: multiset_int > $o,M5: multiset_int] :
( ( P2 @ zero_z3170743180189231877et_int )
=> ( ! [X3: int,M6: multiset_int] :
( ( P2 @ M6 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M6 ) )
=> ( ord_less_eq_int @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_int @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_930_multiset__induct__max,axiom,
! [P2: multiset_num > $o,M5: multiset_num] :
( ( P2 @ zero_z8056838136647266291et_num )
=> ( ! [X3: num,M6: multiset_num] :
( ( P2 @ M6 )
=> ( ! [Xa: num] :
( ( member_num @ Xa @ ( set_mset_num @ M6 ) )
=> ( ord_less_eq_num @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_num @ X3 @ M6 ) ) ) )
=> ( P2 @ M5 ) ) ) ).
% multiset_induct_max
thf(fact_931_size__Diff1__le,axiom,
! [M5: multiset_set_a,X: set_a] : ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M5 ) ) ).
% size_Diff1_le
thf(fact_932_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_933_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_934_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_935_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_936_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_937_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_938_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_939_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_940_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_941_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_942_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_943_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_944_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_945_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_946_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > real,M: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_947_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_948_size__Diff2__less,axiom,
! [X: a,M5: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M5 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M5 ) ) ) ) ).
% size_Diff2_less
thf(fact_949_size__Diff2__less,axiom,
! [X: real,M5: multiset_real,Y: real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ( member_real @ Y @ ( set_mset_real @ M5 ) )
=> ( ord_less_nat @ ( size_s3818332516149715216t_real @ ( minus_3865385036109388885t_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) ) ) @ ( size_s3818332516149715216t_real @ M5 ) ) ) ) ).
% size_Diff2_less
thf(fact_950_size__Diff2__less,axiom,
! [X: nat,M5: multiset_nat,Y: nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ( member_nat @ Y @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M5 ) ) ) ) ).
% size_Diff2_less
thf(fact_951_size__Diff2__less,axiom,
! [X: set_a,M5: multiset_set_a,Y: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ( member_set_a @ Y @ ( set_mset_set_a @ M5 ) )
=> ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M5 ) ) ) ) ).
% size_Diff2_less
thf(fact_952_size__Diff1__less,axiom,
! [X: a,M5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M5 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M5 ) ) ) ).
% size_Diff1_less
thf(fact_953_size__Diff1__less,axiom,
! [X: real,M5: multiset_real] :
( ( member_real @ X @ ( set_mset_real @ M5 ) )
=> ( ord_less_nat @ ( size_s3818332516149715216t_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) ) @ ( size_s3818332516149715216t_real @ M5 ) ) ) ).
% size_Diff1_less
thf(fact_954_size__Diff1__less,axiom,
! [X: nat,M5: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M5 ) ) ) ).
% size_Diff1_less
thf(fact_955_size__Diff1__less,axiom,
! [X: set_a,M5: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M5 ) )
=> ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M5 ) ) ) ).
% size_Diff1_less
thf(fact_956_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_957_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_958_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_959_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ 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 @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_960_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_961_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ 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 @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_962_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_963_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_964_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_965_in__diffD,axiom,
! [A: a,M5: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N4 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ).
% in_diffD
thf(fact_966_in__diffD,axiom,
! [A: real,M5: multiset_real,N4: multiset_real] :
( ( member_real @ A @ ( set_mset_real @ ( minus_3865385036109388885t_real @ M5 @ N4 ) ) )
=> ( member_real @ A @ ( set_mset_real @ M5 ) ) ) ).
% in_diffD
thf(fact_967_in__diffD,axiom,
! [A: nat,M5: multiset_nat,N4: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ N4 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ M5 ) ) ) ).
% in_diffD
thf(fact_968_in__diffD,axiom,
! [A: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ N4 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ M5 ) ) ) ).
% in_diffD
thf(fact_969_ex__mset,axiom,
! [X5: multiset_set_a] :
? [Xs2: list_set_a] :
( ( mset_set_a @ Xs2 )
= X5 ) ).
% ex_mset
thf(fact_970_mset__eq__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_971_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A2 @ A2 )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_972_diff__empty,axiom,
! [M5: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ M5 @ zero_z5079479921072680283_set_a )
= M5 )
& ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M5 )
= zero_z5079479921072680283_set_a ) ) ).
% diff_empty
thf(fact_973_sum__power__add,axiom,
! [X: int,M: nat,I5: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_974_sum__power__add,axiom,
! [X: real,M: nat,I5: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_975_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_976_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_977_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_978_multiset__nonemptyE,axiom,
! [A2: multiset_real] :
( ( A2 != zero_z8811559133707751557t_real )
=> ~ ! [X3: real] :
~ ( member_real @ X3 @ ( set_mset_real @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_979_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_980_multiset__nonemptyE,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
=> ~ ! [X3: set_a] :
~ ( member_set_a @ X3 @ ( set_mset_set_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_981_diff__size__le__size__Diff,axiom,
! [M5: multiset_set_a,M7: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M5 ) @ ( size_s6566526139600085008_set_a @ M7 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M5 @ M7 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_982_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_983_nth__mem__mset,axiom,
! [I: nat,Ls: list_real] :
( ( ord_less_nat @ I @ ( size_size_list_real @ Ls ) )
=> ( member_real @ ( nth_real @ Ls @ I ) @ ( set_mset_real @ ( mset_real @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_984_nth__mem__mset,axiom,
! [I: nat,Ls: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
=> ( member_nat @ ( nth_nat @ Ls @ I ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_985_nth__mem__mset,axiom,
! [I: nat,Ls: list_set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
=> ( member_set_a @ ( nth_set_a @ Ls @ I ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_986_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_987_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_988_nonempty__has__size,axiom,
! [S2: multiset_set_a] :
( ( S2 != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_989_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_990_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_991_mset__lt__single__right__iff,axiom,
! [M5: multiset_nat,Y: nat] :
( ( ord_le5777773500796000884et_nat @ M5 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ X2 @ Y ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_992_mset__lt__single__right__iff,axiom,
! [M5: multiset_num,Y: num] :
( ( ord_le6486017437744838590et_num @ M5 @ ( add_mset_num @ Y @ zero_z8056838136647266291et_num ) )
= ( ! [X2: num] :
( ( member_num @ X2 @ ( set_mset_num @ M5 ) )
=> ( ord_less_num @ X2 @ Y ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_993_mset__lt__single__right__iff,axiom,
! [M5: multiset_real,Y: real] :
( ( ord_le7573655249420395216t_real @ M5 @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_mset_real @ M5 ) )
=> ( ord_less_real @ X2 @ Y ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_994_mset__lt__single__right__iff,axiom,
! [M5: multiset_int,Y: int] :
( ( ord_le1599922481286804176et_int @ M5 @ ( add_mset_int @ Y @ zero_z3170743180189231877et_int ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_mset_int @ M5 ) )
=> ( ord_less_int @ X2 @ Y ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_995_mset__le__single__right__iff,axiom,
! [M5: multiset_nat,Y: nat] :
( ( ord_le6602235886369790592et_nat @ M5 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ( M5
= ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
| ! [X2: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ X2 @ Y ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_996_mset__le__single__right__iff,axiom,
! [M5: multiset_num,Y: num] :
( ( ord_le7310479823318628298et_num @ M5 @ ( add_mset_num @ Y @ zero_z8056838136647266291et_num ) )
= ( ( M5
= ( add_mset_num @ Y @ zero_z8056838136647266291et_num ) )
| ! [X2: num] :
( ( member_num @ X2 @ ( set_mset_num @ M5 ) )
=> ( ord_less_num @ X2 @ Y ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_997_mset__le__single__right__iff,axiom,
! [M5: multiset_real,Y: real] :
( ( ord_le2426415917361421532t_real @ M5 @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
= ( ( M5
= ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
| ! [X2: real] :
( ( member_real @ X2 @ ( set_mset_real @ M5 ) )
=> ( ord_less_real @ X2 @ Y ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_998_mset__le__single__right__iff,axiom,
! [M5: multiset_int,Y: int] :
( ( ord_le2424384866860593884et_int @ M5 @ ( add_mset_int @ Y @ zero_z3170743180189231877et_int ) )
= ( ( M5
= ( add_mset_int @ Y @ zero_z3170743180189231877et_int ) )
| ! [X2: int] :
( ( member_int @ X2 @ ( set_mset_int @ M5 ) )
=> ( ord_less_int @ X2 @ Y ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_999_index__remove1__mset__ne,axiom,
! [X: a,Xs: list_a,Y: a,J1: nat] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
=> ( ( member_a @ Y @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ ( mset_a @ Xs ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) )
=> ( ( ( nth_a @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_a @ Xs ) )
=> ~ ! [J23: nat] :
( ( ( nth_a @ Xs @ J23 )
= Y )
=> ( ( ord_less_nat @ J23 @ ( size_size_list_a @ Xs ) )
=> ( J1 = J23 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_1000_index__remove1__mset__ne,axiom,
! [X: real,Xs: list_real,Y: real,J1: nat] :
( ( member_real @ X @ ( set_mset_real @ ( mset_real @ Xs ) ) )
=> ( ( member_real @ Y @ ( set_mset_real @ ( minus_3865385036109388885t_real @ ( mset_real @ Xs ) @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) ) )
=> ( ( ( nth_real @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_real @ Xs ) )
=> ~ ! [J23: nat] :
( ( ( nth_real @ Xs @ J23 )
= Y )
=> ( ( ord_less_nat @ J23 @ ( size_size_list_real @ Xs ) )
=> ( J1 = J23 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_1001_index__remove1__mset__ne,axiom,
! [X: nat,Xs: list_nat,Y: nat,J1: nat] :
( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
=> ( ( member_nat @ Y @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ ( mset_nat @ Xs ) @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( ( nth_nat @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_nat @ Xs ) )
=> ~ ! [J23: nat] :
( ( ( nth_nat @ Xs @ J23 )
= Y )
=> ( ( ord_less_nat @ J23 @ ( size_size_list_nat @ Xs ) )
=> ( J1 = J23 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_1002_index__remove1__mset__ne,axiom,
! [X: set_a,Xs: list_set_a,Y: set_a,J1: nat] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
=> ( ( member_set_a @ Y @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ Xs ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( ( nth_set_a @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_set_a @ Xs ) )
=> ~ ! [J23: nat] :
( ( ( nth_set_a @ Xs @ J23 )
= Y )
=> ( ( ord_less_nat @ J23 @ ( size_size_list_set_a @ Xs ) )
=> ( J1 = J23 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_1003_const__intersect,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B1 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( design7842873109100088828mber_a @ B1 @ B22 )
= m ) ) ) ).
% const_intersect
thf(fact_1004_remove1__single__empty__iff,axiom,
! [L3: set_a,L4: set_a] :
( ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= zero_z5079479921072680283_set_a )
= ( L4 = L3 ) ) ).
% remove1_single_empty_iff
thf(fact_1005_inter__num__lt__block__size__strict_I2_J,axiom,
! [Bl12: set_a,Bl22: set_a] :
( ( member_set_a @ Bl12 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( Bl12 != Bl22 )
=> ( ( ( finite_card_a @ Bl12 )
= ( finite_card_a @ Bl22 ) )
=> ( ord_less_nat @ ( design7842873109100088828mber_a @ Bl12 @ Bl22 ) @ ( finite_card_a @ Bl22 ) ) ) ) ) ) ).
% inter_num_lt_block_size_strict(2)
thf(fact_1006_inter__num__lt__block__size__strict_I1_J,axiom,
! [Bl12: set_a,Bl22: set_a] :
( ( member_set_a @ Bl12 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( Bl12 != Bl22 )
=> ( ( ( finite_card_a @ Bl12 )
= ( finite_card_a @ Bl22 ) )
=> ( ord_less_nat @ ( design7842873109100088828mber_a @ Bl12 @ Bl22 ) @ ( finite_card_a @ Bl12 ) ) ) ) ) ) ).
% inter_num_lt_block_size_strict(1)
thf(fact_1007_indexed__const__intersect,axiom,
! [J1: nat,J22: nat] :
( ( ord_less_nat @ J1 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ J22 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( J1 != J22 )
=> ( ( design7842873109100088828mber_a @ ( nth_set_a @ b_s @ J1 ) @ ( nth_set_a @ b_s @ J22 ) )
= m ) ) ) ) ).
% indexed_const_intersect
thf(fact_1008_remove__diff__multiset,axiom,
! [X13: a,A2: multiset_a,B3: multiset_a] :
( ~ ( member_a @ X13 @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X13 @ B3 ) )
= ( minus_3765977307040488491iset_a @ A2 @ B3 ) ) ) ).
% remove_diff_multiset
thf(fact_1009_remove__diff__multiset,axiom,
! [X13: real,A2: multiset_real,B3: multiset_real] :
( ~ ( member_real @ X13 @ ( set_mset_real @ A2 ) )
=> ( ( minus_3865385036109388885t_real @ A2 @ ( add_mset_real @ X13 @ B3 ) )
= ( minus_3865385036109388885t_real @ A2 @ B3 ) ) ) ).
% remove_diff_multiset
thf(fact_1010_remove__diff__multiset,axiom,
! [X13: nat,A2: multiset_nat,B3: multiset_nat] :
( ~ ( member_nat @ X13 @ ( set_mset_nat @ A2 ) )
=> ( ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X13 @ B3 ) )
= ( minus_8522176038001411705et_nat @ A2 @ B3 ) ) ) ).
% remove_diff_multiset
thf(fact_1011_remove__diff__multiset,axiom,
! [X13: set_a,A2: multiset_set_a,B3: multiset_set_a] :
( ~ ( member_set_a @ X13 @ ( set_mset_set_a @ A2 ) )
=> ( ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X13 @ B3 ) )
= ( minus_706656509937749387_set_a @ A2 @ B3 ) ) ) ).
% remove_diff_multiset
thf(fact_1012_mset__lt__single__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
= ( ord_less_set_a @ X @ Y ) ) ).
% mset_lt_single_iff
thf(fact_1013_mset__lt__single__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ord_less_nat @ X @ Y ) ) ).
% mset_lt_single_iff
thf(fact_1014_mset__lt__single__iff,axiom,
! [X: num,Y: num] :
( ( ord_le6486017437744838590et_num @ ( add_mset_num @ X @ zero_z8056838136647266291et_num ) @ ( add_mset_num @ Y @ zero_z8056838136647266291et_num ) )
= ( ord_less_num @ X @ Y ) ) ).
% mset_lt_single_iff
thf(fact_1015_mset__lt__single__iff,axiom,
! [X: real,Y: real] :
( ( ord_le7573655249420395216t_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
= ( ord_less_real @ X @ Y ) ) ).
% mset_lt_single_iff
thf(fact_1016_mset__lt__single__iff,axiom,
! [X: int,Y: int] :
( ( ord_le1599922481286804176et_int @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ Y @ zero_z3170743180189231877et_int ) )
= ( ord_less_int @ X @ Y ) ) ).
% mset_lt_single_iff
thf(fact_1017_mset__le__single__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le7905258569527593284_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1018_mset__le__single__iff,axiom,
! [X: real,Y: real] :
( ( ord_le2426415917361421532t_real @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) @ ( add_mset_real @ Y @ zero_z8811559133707751557t_real ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1019_mset__le__single__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_le6602235886369790592et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1020_mset__le__single__iff,axiom,
! [X: int,Y: int] :
( ( ord_le2424384866860593884et_int @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) @ ( add_mset_int @ Y @ zero_z3170743180189231877et_int ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1021_mset__le__single__iff,axiom,
! [X: num,Y: num] :
( ( ord_le7310479823318628298et_num @ ( add_mset_num @ X @ zero_z8056838136647266291et_num ) @ ( add_mset_num @ Y @ zero_z8056838136647266291et_num ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1022_mset__le__single__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le4034546139768944438et_nat @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) @ ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% mset_le_single_iff
thf(fact_1023_list__of__mset__exi,axiom,
! [M: multiset_set_a] :
? [L5: list_set_a] :
( M
= ( mset_set_a @ L5 ) ) ).
% list_of_mset_exi
thf(fact_1024_le__multiset__empty__left,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ M5 ) ) ).
% le_multiset_empty_left
thf(fact_1025_le__multiset__empty__right,axiom,
! [M5: multiset_set_a] :
~ ( ord_le5765082015083327056_set_a @ M5 @ zero_z5079479921072680283_set_a ) ).
% le_multiset_empty_right
thf(fact_1026_less__eq__multiset__empty__left,axiom,
! [M5: multiset_set_a] : ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ M5 ) ).
% less_eq_multiset_empty_left
thf(fact_1027_less__eq__multiset__empty__right,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ~ ( ord_le7905258569527593284_set_a @ M5 @ zero_z5079479921072680283_set_a ) ) ).
% less_eq_multiset_empty_right
thf(fact_1028_le__multiset__plus__right__nonempty,axiom,
! [N4: multiset_set_a,M5: multiset_set_a] :
( ( N4 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ M5 @ ( plus_p2331992037799027419_set_a @ M5 @ N4 ) ) ) ).
% le_multiset_plus_right_nonempty
thf(fact_1029_le__multiset__plus__left__nonempty,axiom,
! [M5: multiset_set_a,N4: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ N4 @ ( plus_p2331992037799027419_set_a @ M5 @ N4 ) ) ) ).
% le_multiset_plus_left_nonempty
thf(fact_1030_le__multiset__right__total,axiom,
! [M5: multiset_set_a,X: set_a] : ( ord_le5765082015083327056_set_a @ M5 @ ( add_mset_set_a @ X @ M5 ) ) ).
% le_multiset_right_total
thf(fact_1031_sum__reorder__triple,axiom,
! [G: nat > nat > nat > real,C2: set_nat,B3: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [L6: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( G @ L6 @ I4 ) @ C2 )
@ B3 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [L6: nat] : ( G @ L6 @ I4 @ J2 )
@ A2 )
@ C2 )
@ B3 ) ) ).
% sum_reorder_triple
thf(fact_1032_sum__reorder__triple,axiom,
! [G: nat > nat > nat > nat,C2: set_nat,B3: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [L6: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ L6 @ I4 ) @ C2 )
@ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [L6: nat] : ( G @ L6 @ I4 @ J2 )
@ A2 )
@ C2 )
@ B3 ) ) ).
% sum_reorder_triple
thf(fact_1033_ex__gt__imp__less__multiset,axiom,
! [N4: multiset_set_a,M5: multiset_set_a] :
( ? [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_mset_set_a @ N4 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ M5 ) )
=> ( ord_less_set_a @ X3 @ Y3 ) ) )
=> ( ord_le5765082015083327056_set_a @ M5 @ N4 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1034_ex__gt__imp__less__multiset,axiom,
! [N4: multiset_nat,M5: multiset_nat] :
( ? [Y3: nat] :
( ( member_nat @ Y3 @ ( set_mset_nat @ N4 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ X3 @ Y3 ) ) )
=> ( ord_le5777773500796000884et_nat @ M5 @ N4 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1035_ex__gt__imp__less__multiset,axiom,
! [N4: multiset_num,M5: multiset_num] :
( ? [Y3: num] :
( ( member_num @ Y3 @ ( set_mset_num @ N4 ) )
& ! [X3: num] :
( ( member_num @ X3 @ ( set_mset_num @ M5 ) )
=> ( ord_less_num @ X3 @ Y3 ) ) )
=> ( ord_le6486017437744838590et_num @ M5 @ N4 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1036_ex__gt__imp__less__multiset,axiom,
! [N4: multiset_real,M5: multiset_real] :
( ? [Y3: real] :
( ( member_real @ Y3 @ ( set_mset_real @ N4 ) )
& ! [X3: real] :
( ( member_real @ X3 @ ( set_mset_real @ M5 ) )
=> ( ord_less_real @ X3 @ Y3 ) ) )
=> ( ord_le7573655249420395216t_real @ M5 @ N4 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1037_ex__gt__imp__less__multiset,axiom,
! [N4: multiset_int,M5: multiset_int] :
( ? [Y3: int] :
( ( member_int @ Y3 @ ( set_mset_int @ N4 ) )
& ! [X3: int] :
( ( member_int @ X3 @ ( set_mset_int @ M5 ) )
=> ( ord_less_int @ X3 @ Y3 ) ) )
=> ( ord_le1599922481286804176et_int @ M5 @ N4 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1038_double__sum__mult__hom,axiom,
! [K: real,F: nat > nat > real,G: nat > set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ K @ ( F @ I4 @ J2 ) )
@ ( G @ I4 ) )
@ A2 )
= ( times_times_real @ K
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( F @ I4 ) @ ( G @ I4 ) )
@ A2 ) ) ) ).
% double_sum_mult_hom
thf(fact_1039_double__sum__split__case2,axiom,
! [G: a > a > real,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I4: a] : ( groups2740460157737275248a_real @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_real
@ ( groups2740460157737275248a_real
@ ^ [I4: a] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups2740460157737275248a_real
@ ^ [I4: a] :
( groups2740460157737275248a_real @ ( G @ I4 )
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1040_double__sum__split__case2,axiom,
! [G: real > real > real,A2: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I4: real] : ( groups8097168146408367636l_real @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_real
@ ( groups8097168146408367636l_real
@ ^ [I4: real] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups8097168146408367636l_real
@ ^ [I4: real] :
( groups8097168146408367636l_real @ ( G @ I4 )
@ ( collect_real
@ ^ [A4: real] :
( ( member_real @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1041_double__sum__split__case2,axiom,
! [G: a > a > nat,A2: set_a] :
( ( groups6334556678337121940_a_nat
@ ^ [I4: a] : ( groups6334556678337121940_a_nat @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_nat
@ ( groups6334556678337121940_a_nat
@ ^ [I4: a] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups6334556678337121940_a_nat
@ ^ [I4: a] :
( groups6334556678337121940_a_nat @ ( G @ I4 )
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1042_double__sum__split__case2,axiom,
! [G: real > real > nat,A2: set_real] :
( ( groups1935376822645274424al_nat
@ ^ [I4: real] : ( groups1935376822645274424al_nat @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_nat
@ ( groups1935376822645274424al_nat
@ ^ [I4: real] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups1935376822645274424al_nat
@ ^ [I4: real] :
( groups1935376822645274424al_nat @ ( G @ I4 )
@ ( collect_real
@ ^ [A4: real] :
( ( member_real @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1043_double__sum__split__case2,axiom,
! [G: a > a > int,A2: set_a] :
( ( groups6332066207828071664_a_int
@ ^ [I4: a] : ( groups6332066207828071664_a_int @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_int
@ ( groups6332066207828071664_a_int
@ ^ [I4: a] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups6332066207828071664_a_int
@ ^ [I4: a] :
( groups6332066207828071664_a_int @ ( G @ I4 )
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1044_double__sum__split__case2,axiom,
! [G: real > real > int,A2: set_real] :
( ( groups1932886352136224148al_int
@ ^ [I4: real] : ( groups1932886352136224148al_int @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_int
@ ( groups1932886352136224148al_int
@ ^ [I4: real] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups1932886352136224148al_int
@ ^ [I4: real] :
( groups1932886352136224148al_int @ ( G @ I4 )
@ ( collect_real
@ ^ [A4: real] :
( ( member_real @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1045_double__sum__split__case2,axiom,
! [G: nat > nat > int,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( groups3539618377306564664at_int @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] :
( groups3539618377306564664at_int @ ( G @ I4 )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1046_double__sum__split__case2,axiom,
! [G: nat > nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real @ ( G @ I4 )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1047_double__sum__split__case2,axiom,
! [G: nat > nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_nat
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat @ ( G @ I4 )
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1048_double__sum__split__case2,axiom,
! [G: set_a > set_a > real,A2: set_set_a] :
( ( groups9174420418583655632a_real
@ ^ [I4: set_a] : ( groups9174420418583655632a_real @ ( G @ I4 ) @ A2 )
@ A2 )
= ( plus_plus_real
@ ( groups9174420418583655632a_real
@ ^ [I4: set_a] : ( G @ I4 @ I4 )
@ A2 )
@ ( groups9174420418583655632a_real
@ ^ [I4: set_a] :
( groups9174420418583655632a_real @ ( G @ I4 )
@ ( collect_set_a
@ ^ [A4: set_a] :
( ( member_set_a @ A4 @ A2 )
& ( A4 != I4 ) ) ) )
@ A2 ) ) ) ).
% double_sum_split_case2
thf(fact_1049_less__multiset__doubletons,axiom,
! [Y: set_a,T: set_a,S: set_a,X: set_a] :
( ( ( ord_less_set_a @ Y @ T )
| ( ord_less_set_a @ Y @ S ) )
=> ( ( ( ord_less_set_a @ X @ T )
| ( ord_less_set_a @ X @ S ) )
=> ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ Y @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ T @ ( add_mset_set_a @ S @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_1050_less__multiset__doubletons,axiom,
! [Y: nat,T: nat,S: nat,X: nat] :
( ( ( ord_less_nat @ Y @ T )
| ( ord_less_nat @ Y @ S ) )
=> ( ( ( ord_less_nat @ X @ T )
| ( ord_less_nat @ X @ S ) )
=> ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ Y @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ T @ ( add_mset_nat @ S @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_1051_less__multiset__doubletons,axiom,
! [Y: num,T: num,S: num,X: num] :
( ( ( ord_less_num @ Y @ T )
| ( ord_less_num @ Y @ S ) )
=> ( ( ( ord_less_num @ X @ T )
| ( ord_less_num @ X @ S ) )
=> ( ord_le6486017437744838590et_num @ ( add_mset_num @ Y @ ( add_mset_num @ X @ zero_z8056838136647266291et_num ) ) @ ( add_mset_num @ T @ ( add_mset_num @ S @ zero_z8056838136647266291et_num ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_1052_less__multiset__doubletons,axiom,
! [Y: real,T: real,S: real,X: real] :
( ( ( ord_less_real @ Y @ T )
| ( ord_less_real @ Y @ S ) )
=> ( ( ( ord_less_real @ X @ T )
| ( ord_less_real @ X @ S ) )
=> ( ord_le7573655249420395216t_real @ ( add_mset_real @ Y @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) @ ( add_mset_real @ T @ ( add_mset_real @ S @ zero_z8811559133707751557t_real ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_1053_less__multiset__doubletons,axiom,
! [Y: int,T: int,S: int,X: int] :
( ( ( ord_less_int @ Y @ T )
| ( ord_less_int @ Y @ S ) )
=> ( ( ( ord_less_int @ X @ T )
| ( ord_less_int @ X @ S ) )
=> ( ord_le1599922481286804176et_int @ ( add_mset_int @ Y @ ( add_mset_int @ X @ zero_z3170743180189231877et_int ) ) @ ( add_mset_int @ T @ ( add_mset_int @ S @ zero_z3170743180189231877et_int ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_1054_add__mset__less__imp__less__remove1__mset,axiom,
! [X: set_a,M5: multiset_set_a,N4: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ X @ M5 ) @ N4 )
=> ( ord_le5765082015083327056_set_a @ M5 @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ).
% add_mset_less_imp_less_remove1_mset
thf(fact_1055_remove1__mset__add__mset__If,axiom,
! [L4: set_a,L3: set_a,C2: multiset_set_a] :
( ( ( L4 = L3 )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ C2 ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= C2 ) )
& ( ( L4 != L3 )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ C2 ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ L3 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% remove1_mset_add_mset_If
thf(fact_1056_in__remove1__mset__neq,axiom,
! [A: a,B: a,C2: multiset_a] :
( ( A != B )
=> ( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ C2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_1057_in__remove1__mset__neq,axiom,
! [A: real,B: real,C2: multiset_real] :
( ( A != B )
=> ( ( member_real @ A @ ( set_mset_real @ ( minus_3865385036109388885t_real @ C2 @ ( add_mset_real @ B @ zero_z8811559133707751557t_real ) ) ) )
= ( member_real @ A @ ( set_mset_real @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_1058_in__remove1__mset__neq,axiom,
! [A: nat,B: nat,C2: multiset_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ C2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A @ ( set_mset_nat @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_1059_in__remove1__mset__neq,axiom,
! [A: set_a,B: set_a,C2: multiset_set_a] :
( ( A != B )
=> ( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
= ( member_set_a @ A @ ( set_mset_set_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_1060_add__mset__eq__add__mset,axiom,
! [A: a,M5: multiset_a,B: a,M7: multiset_a] :
( ( ( add_mset_a @ A @ M5 )
= ( add_mset_a @ B @ M7 ) )
= ( ( ( A = B )
& ( M5 = M7 ) )
| ( ( A != B )
& ( member_a @ B @ ( set_mset_a @ M5 ) )
& ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= M7 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_1061_add__mset__eq__add__mset,axiom,
! [A: real,M5: multiset_real,B: real,M7: multiset_real] :
( ( ( add_mset_real @ A @ M5 )
= ( add_mset_real @ B @ M7 ) )
= ( ( ( A = B )
& ( M5 = M7 ) )
| ( ( A != B )
& ( member_real @ B @ ( set_mset_real @ M5 ) )
& ( ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ B @ zero_z8811559133707751557t_real ) ) )
= M7 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_1062_add__mset__eq__add__mset,axiom,
! [A: nat,M5: multiset_nat,B: nat,M7: multiset_nat] :
( ( ( add_mset_nat @ A @ M5 )
= ( add_mset_nat @ B @ M7 ) )
= ( ( ( A = B )
& ( M5 = M7 ) )
| ( ( A != B )
& ( member_nat @ B @ ( set_mset_nat @ M5 ) )
& ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= M7 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_1063_add__mset__eq__add__mset,axiom,
! [A: set_a,M5: multiset_set_a,B: set_a,M7: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M5 )
= ( add_mset_set_a @ B @ M7 ) )
= ( ( ( A = B )
& ( M5 = M7 ) )
| ( ( A != B )
& ( member_set_a @ B @ ( set_mset_set_a @ M5 ) )
& ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= M7 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_1064_add__mset__eq__add__mset__ne,axiom,
! [A: a,B: a,A2: multiset_a,B3: multiset_a] :
( ( A != B )
=> ( ( ( add_mset_a @ A @ A2 )
= ( add_mset_a @ B @ B3 ) )
= ( ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( member_a @ B @ ( set_mset_a @ A2 ) )
& ( A2
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_1065_add__mset__eq__add__mset__ne,axiom,
! [A: real,B: real,A2: multiset_real,B3: multiset_real] :
( ( A != B )
=> ( ( ( add_mset_real @ A @ A2 )
= ( add_mset_real @ B @ B3 ) )
= ( ( member_real @ A @ ( set_mset_real @ B3 ) )
& ( member_real @ B @ ( set_mset_real @ A2 ) )
& ( A2
= ( add_mset_real @ B @ ( minus_3865385036109388885t_real @ B3 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_1066_add__mset__eq__add__mset__ne,axiom,
! [A: nat,B: nat,A2: multiset_nat,B3: multiset_nat] :
( ( A != B )
=> ( ( ( add_mset_nat @ A @ A2 )
= ( add_mset_nat @ B @ B3 ) )
= ( ( member_nat @ A @ ( set_mset_nat @ B3 ) )
& ( member_nat @ B @ ( set_mset_nat @ A2 ) )
& ( A2
= ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_1067_add__mset__eq__add__mset__ne,axiom,
! [A: set_a,B: set_a,A2: multiset_set_a,B3: multiset_set_a] :
( ( A != B )
=> ( ( ( add_mset_set_a @ A @ A2 )
= ( add_mset_set_a @ B @ B3 ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ B3 ) )
& ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
& ( A2
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_1068_id__remove__1__mset__iff__notin,axiom,
! [M5: multiset_a,A: a] :
( ( M5
= ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( ~ ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_1069_id__remove__1__mset__iff__notin,axiom,
! [M5: multiset_real,A: real] :
( ( M5
= ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) ) )
= ( ~ ( member_real @ A @ ( set_mset_real @ M5 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_1070_id__remove__1__mset__iff__notin,axiom,
! [M5: multiset_nat,A: nat] :
( ( M5
= ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( ~ ( member_nat @ A @ ( set_mset_nat @ M5 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_1071_id__remove__1__mset__iff__notin,axiom,
! [M5: multiset_set_a,A: set_a] :
( ( M5
= ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( ~ ( member_set_a @ A @ ( set_mset_set_a @ M5 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_1072_remove__1__mset__id__iff__notin,axiom,
! [M5: multiset_a,A: a] :
( ( ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= M5 )
= ( ~ ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_1073_remove__1__mset__id__iff__notin,axiom,
! [M5: multiset_real,A: real] :
( ( ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ A @ zero_z8811559133707751557t_real ) )
= M5 )
= ( ~ ( member_real @ A @ ( set_mset_real @ M5 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_1074_remove__1__mset__id__iff__notin,axiom,
! [M5: multiset_nat,A: nat] :
( ( ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= M5 )
= ( ~ ( member_nat @ A @ ( set_mset_nat @ M5 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_1075_remove__1__mset__id__iff__notin,axiom,
! [M5: multiset_set_a,A: set_a] :
( ( ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= M5 )
= ( ~ ( member_set_a @ A @ ( set_mset_set_a @ M5 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_1076_add__mset__remove__trivial__iff,axiom,
! [N4: multiset_a,A: a,B: a] :
( ( N4
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
= ( ( member_a @ A @ ( set_mset_a @ N4 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_1077_add__mset__remove__trivial__iff,axiom,
! [N4: multiset_real,A: real,B: real] :
( ( N4
= ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ N4 @ ( add_mset_real @ B @ zero_z8811559133707751557t_real ) ) ) )
= ( ( member_real @ A @ ( set_mset_real @ N4 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_1078_add__mset__remove__trivial__iff,axiom,
! [N4: multiset_nat,A: nat,B: nat] :
( ( N4
= ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ N4 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_1079_add__mset__remove__trivial__iff,axiom,
! [N4: multiset_set_a,A: set_a,B: set_a] :
( ( N4
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ N4 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_1080_trivial__add__mset__remove__iff,axiom,
! [A: a,N4: multiset_a,B: a] :
( ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= N4 )
= ( ( member_a @ A @ ( set_mset_a @ N4 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_1081_trivial__add__mset__remove__iff,axiom,
! [A: real,N4: multiset_real,B: real] :
( ( ( add_mset_real @ A @ ( minus_3865385036109388885t_real @ N4 @ ( add_mset_real @ B @ zero_z8811559133707751557t_real ) ) )
= N4 )
= ( ( member_real @ A @ ( set_mset_real @ N4 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_1082_trivial__add__mset__remove__iff,axiom,
! [A: nat,N4: multiset_nat,B: nat] :
( ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= N4 )
= ( ( member_nat @ A @ ( set_mset_nat @ N4 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_1083_trivial__add__mset__remove__iff,axiom,
! [A: set_a,N4: multiset_set_a,B: set_a] :
( ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N4 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= N4 )
= ( ( member_set_a @ A @ ( set_mset_set_a @ N4 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_1084_remove1__mset__eqE,axiom,
! [X1: multiset_a,L4: a,M5: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ X1 @ ( add_mset_a @ L4 @ zero_zero_multiset_a ) )
= M5 )
=> ( ( ( member_a @ L4 @ ( set_mset_a @ X1 ) )
=> ( X1
!= ( plus_plus_multiset_a @ M5 @ ( add_mset_a @ L4 @ zero_zero_multiset_a ) ) ) )
=> ~ ( ~ ( member_a @ L4 @ ( set_mset_a @ X1 ) )
=> ( X1 != M5 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1085_remove1__mset__eqE,axiom,
! [X1: multiset_real,L4: real,M5: multiset_real] :
( ( ( minus_3865385036109388885t_real @ X1 @ ( add_mset_real @ L4 @ zero_z8811559133707751557t_real ) )
= M5 )
=> ( ( ( member_real @ L4 @ ( set_mset_real @ X1 ) )
=> ( X1
!= ( plus_p8661369373666671365t_real @ M5 @ ( add_mset_real @ L4 @ zero_z8811559133707751557t_real ) ) ) )
=> ~ ( ~ ( member_real @ L4 @ ( set_mset_real @ X1 ) )
=> ( X1 != M5 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1086_remove1__mset__eqE,axiom,
! [X1: multiset_nat,L4: nat,M5: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ X1 @ ( add_mset_nat @ L4 @ zero_z7348594199698428585et_nat ) )
= M5 )
=> ( ( ( member_nat @ L4 @ ( set_mset_nat @ X1 ) )
=> ( X1
!= ( plus_p6334493942879108393et_nat @ M5 @ ( add_mset_nat @ L4 @ zero_z7348594199698428585et_nat ) ) ) )
=> ~ ( ~ ( member_nat @ L4 @ ( set_mset_nat @ X1 ) )
=> ( X1 != M5 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1087_remove1__mset__eqE,axiom,
! [X1: multiset_set_a,L4: set_a,M5: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ X1 @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= M5 )
=> ( ( ( member_set_a @ L4 @ ( set_mset_set_a @ X1 ) )
=> ( X1
!= ( plus_p2331992037799027419_set_a @ M5 @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) ) ) )
=> ~ ( ~ ( member_set_a @ L4 @ ( set_mset_set_a @ X1 ) )
=> ( X1 != M5 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1088_in__mset__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_1089_in__mset__conv__nth,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_mset_real @ ( mset_real @ Xs ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I4 )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_1090_in__mset__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_1091_in__mset__conv__nth,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_1092_size__mset__remove1__mset__le__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M5 ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_1093_size__mset__remove1__mset__le__iff,axiom,
! [M5: multiset_real,X: real] :
( ( ord_less_nat @ ( size_s3818332516149715216t_real @ ( minus_3865385036109388885t_real @ M5 @ ( add_mset_real @ X @ zero_z8811559133707751557t_real ) ) ) @ ( size_s3818332516149715216t_real @ M5 ) )
= ( member_real @ X @ ( set_mset_real @ M5 ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_1094_size__mset__remove1__mset__le__iff,axiom,
! [M5: multiset_nat,X: nat] :
( ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M5 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M5 ) )
= ( member_nat @ X @ ( set_mset_nat @ M5 ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_1095_size__mset__remove1__mset__le__iff,axiom,
! [M5: multiset_set_a,X: set_a] :
( ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M5 ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M5 ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_1096_intersect__num__in__set,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B1 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( member_nat @ ( design7842873109100088828mber_a @ B1 @ B22 ) @ ( design3761797438660848528bers_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% intersect_num_in_set
thf(fact_1097_obtain__blocks__intersect__num,axiom,
! [N: nat] :
( ( member_nat @ N @ ( design3761797438660848528bers_a @ ( mset_set_a @ b_s ) ) )
=> ? [B12: set_a,B23: set_a] :
( ( member_set_a @ B12 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( member_set_a @ B23 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B12 @ zero_z5079479921072680283_set_a ) ) ) )
& ( ( design7842873109100088828mber_a @ B12 @ B23 )
= N ) ) ) ).
% obtain_blocks_intersect_num
thf(fact_1098_dual__sys_Oblock__complement__size,axiom,
! [B: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( finite_card_nat @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ ( finite_card_nat @ B ) ) ) ) ).
% dual_sys.block_complement_size
thf(fact_1099_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1100_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1101_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1102_dual__sys_Oblock__complement__def,axiom,
! [B: set_nat] :
( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B )
= ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) ) ).
% dual_sys.block_complement_def
thf(fact_1103_dual__sys_Oblock__complement__subset__points,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
=> ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.block_complement_subset_points
thf(fact_1104_dual__sys_Oblock__complement__elem__iff,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ Ps )
=> ~ ( member_nat @ X2 @ Bl2 ) ) ) ) ) ).
% dual_sys.block_complement_elem_iff
thf(fact_1105_dual__sys_Oblock__comp__elem__alt__right,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ Ps )
=> ~ ( member_nat @ X3 @ Bl2 ) )
=> ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) ) ) ) ).
% dual_sys.block_comp_elem_alt_right
thf(fact_1106_dual__sys_Oblock__comp__elem__alt__left,axiom,
! [X: nat,Bl2: set_nat,Ps: set_nat] :
( ( member_nat @ X @ Bl2 )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
=> ~ ( member_nat @ X @ Ps ) ) ) ).
% dual_sys.block_comp_elem_alt_left
thf(fact_1107_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1108_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1109_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1110_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1111_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1112_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1113_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1114_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1115_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1116_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1117_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1118_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1119_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1120_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1121_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1122_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1123_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1124_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1125_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1126_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1127_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1128_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1129_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1130_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1131_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1132_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1133_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1134_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1135_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1136_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1137_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1138_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1139_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1140_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1141_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1142_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1143_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1144_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1145_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_1146_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1147_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1148_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_1149_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1150_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_1151_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1152_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1153_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1154_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1155_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1156_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1157_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1158_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1159_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1160_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1161_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1162_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1163_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1164_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1165_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1166_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1167_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1168_lambda__zero,axiom,
( ( ^ [H3: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_1169_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_1170_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_1171_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1172_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1173_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1174_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1175_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1176_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1177_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1178_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1179_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1180_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1181_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1182_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1183_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1184_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1185_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1186_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1187_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1188_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_1189_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1190_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1191_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1192_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1193_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1194_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1195_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1196_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1197_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1198_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1199_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1200_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1201_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1202_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1203_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1204_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1205_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1206_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1207_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_1208_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1209_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1210_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1211_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1212_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ 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 @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1213_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ 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 @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1214_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ 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 @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1215_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1216_intersection__numbers__def,axiom,
( ( design3761797438660848528bers_a @ ( mset_set_a @ b_s ) )
= ( collect_nat
@ ^ [Uu: nat] :
? [B13: set_a,B24: set_a] :
( ( Uu
= ( design7842873109100088828mber_a @ B13 @ B24 ) )
& ( member_set_a @ B13 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( member_set_a @ B24 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B13 @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% intersection_numbers_def
thf(fact_1217_sys__block__sizes__def,axiom,
( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
= ( collect_nat
@ ^ [Uu: nat] :
? [Bl3: set_a] :
( ( Uu
= ( finite_card_a @ Bl3 ) )
& ( member_set_a @ Bl3 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% sys_block_sizes_def
thf(fact_1218_del__block__def,axiom,
! [B: set_a] :
( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
= ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ).
% del_block_def
thf(fact_1219_dual__sys_Odel__invalid__point,axiom,
! [P: nat] :
( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.del_invalid_point
thf(fact_1220_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1221_delete__invalid__block__eq,axiom,
! [B: set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
= ( mset_set_a @ b_s ) ) ) ).
% delete_invalid_block_eq
thf(fact_1222_del__block__b_I2_J,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% del_block_b(2)
thf(fact_1223_dual__sys_Oadd__delete__point__inv,axiom,
! [P: nat] :
( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design4269233978287968195nt_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ P )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.add_delete_point_inv
thf(fact_1224_dual__sys_Odel__point__order,axiom,
! [P: nat] :
( ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( finite_card_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ one_one_nat ) ) ) ).
% dual_sys.del_point_order
thf(fact_1225_del__block__b_I1_J,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_1226_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_1227_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_1228_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1229_dual__sys_Oadd__existing__point,axiom,
! [P: nat] :
( ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.add_existing_point
thf(fact_1230_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1231_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_1232_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1233_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1234_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1235_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1236_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1237_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1238_sum__power2,axiom,
! [K: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_1239_n__inter__num__zero,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ).
% n_inter_num_zero
thf(fact_1240_mult__blocks__const__inter,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( m
= ( finite_card_a @ Bl2 ) ) ) ) ) ).
% mult_blocks_const_inter
thf(fact_1241_simple__alt__def__all,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ X4 )
= one_one_nat ) ) ).
% simple_alt_def_all
thf(fact_1242_const__inter__multiplicity__one,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ m @ ( finite_card_a @ Bl2 ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 )
= one_one_nat ) ) ) ).
% const_inter_multiplicity_one
thf(fact_1243_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1244_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1245_simple,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 )
= one_one_nat ) ) ).
% simple
thf(fact_1246_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1247_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1248_int__ge__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1249_int__gr__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1250_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1251_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_1252_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_1253_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1254_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1255_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_1256_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1257_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1258_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1259_int__induct,axiom,
! [P2: int > $o,K: int,I: int] :
( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_1260_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1261_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1262_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1263_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1264_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_1265_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1266_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_1267_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1268_Bolzano,axiom,
! [A: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B2: real,C3: real] :
( ( P2 @ A3 @ B2 )
=> ( ( P2 @ B2 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C3 )
=> ( P2 @ A3 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D4 ) )
=> ( P2 @ A3 @ B2 ) ) ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1269_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( power_power_real @ ( c @ J2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( ring_1_of_int_real @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( finite_card_a @ ( nth_set_a @ b_s @ J2 ) ) ) @ ( semiri1314217659103216013at_int @ m ) ) ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
%------------------------------------------------------------------------------