TPTP Problem File: SLH0470^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Khovanskii_Theorem/0008_Khovanskii/prob_00348_011568__13424324_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1510 ( 541 unt; 234 typ; 0 def)
% Number of atoms : 3883 (1269 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 14252 ( 224 ~; 64 |; 254 &;11917 @)
% ( 0 <=>;1793 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 7 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 1522 (1522 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 206 usr; 19 con; 0-6 aty)
% Number of variables : 3831 ( 324 ^;3423 !; 84 ?;3831 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:14:09.965
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
list_P7940050157051400743st_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
list_P6254988961118846195et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
list_P3660316430366008877_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1703568184450464039_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mtf__a_J_J,type,
list_nat_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__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (206)
thf(sy_c_Countable__Set_Oto__nat__on_001_062_It__Nat__Onat_Mtf__a_J,type,
counta7388732360187139652_nat_a: set_nat_a > ( nat > a ) > nat ).
thf(sy_c_Countable__Set_Oto__nat__on_001t__Nat__Onat,type,
counta4844910239362777137on_nat: set_nat > nat > nat ).
thf(sy_c_Countable__Set_Oto__nat__on_001tf__a,type,
counta3566351752493190365t_on_a: set_a > a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001_062_It__Nat__Onat_Mtf__a_J,type,
commut2316704705022288065_nat_a: set_nat_a > ( nat > a ) > ( nat > a ) > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001t__Nat__Onat,type,
commut810702690453168372fy_nat: set_nat > nat > nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__a,type,
commutative_M_ify_a: set_a > a > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
commut6753747983606973455_a_nat: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > nat > a ) > set_nat > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
commut1274061894236046463at_a_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( a > nat > a ) > set_a > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001t__Nat__Onat,type,
commut1028764413824576968at_nat: set_nat > ( nat > nat > nat ) > nat > ( nat > nat ) > set_nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001tf__a,type,
commut1549887680474846982_nat_a: set_nat > ( nat > nat > nat ) > nat > ( a > nat ) > set_a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
commut5242989786243415821_nat_a: set_a > ( a > a > a ) > a > ( ( nat > a ) > a ) > set_nat_a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001t__Nat__Onat,type,
commut6741328216151336360_a_nat: set_a > ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001tf__a,type,
commut5005951359559292710mp_a_a: set_a > ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_card_nat_a: set_nat_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on_001t__Nat__Onat_001tf__a,type,
finite1071566134745755356_nat_a: set_nat > ( nat > a > a ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofold_001_062_It__Nat__Onat_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
finite8825492827046180360_nat_a: ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat_a > nat > a ).
thf(sy_c_Finite__Set_Ofold_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
finite7774500027257897325_a_nat: ( ( nat > a ) > nat > nat ) > nat > set_nat_a > nat ).
thf(sy_c_Finite__Set_Ofold_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
finite_fold_nat_a_a: ( ( nat > a ) > a > a ) > a > set_nat_a > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
finite6730669110406474827_nat_a: ( nat > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat > nat > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001tf__a,type,
finite_fold_nat_a: ( nat > a > a ) > a > set_nat > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_fold_a_nat_a: ( a > ( nat > a ) > nat > a ) > ( nat > a ) > set_a > nat > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Nat__Onat,type,
finite_fold_a_nat: ( a > nat > nat ) > nat > set_a > nat ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001tf__a,type,
finite_fold_a_a: ( a > a > a ) > a > set_a > a ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
pi_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_Group__Theory_Oabelian__group_001_062_It__Nat__Onat_Mtf__a_J,type,
group_2359011037596659067_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001t__Nat__Onat,type,
group_4000446350026676922up_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_It__Nat__Onat_Mtf__a_J,type,
group_3093379471365697572_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001t__Nat__Onat,type,
group_6791354081887936081id_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_It__Nat__Onat_Mtf__a_J,type,
group_group_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001t__Nat__Onat,type,
group_group_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_It__Nat__Onat_Mtf__a_J,type,
group_inverse_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > a ) > nat > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_It__Nat__Onat_Mtf__a_J,type,
group_645299334525884886_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_It__Nat__Onat_Mtf__a_J,type,
group_subgroup_nat_a: set_nat_a > set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Nat__Onat_J,type,
minus_minus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_1998526526692677103et_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
minus_6559351004396594891_set_a: list_set_a > list_set_a > list_set_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_I_062_It__Nat__Onat_Mtf__a_J_J,type,
minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).
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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p2116291331692525561st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__Nat__Onat_J,type,
plus_plus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__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__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Real__Oreal,type,
groups6723090944982001619t_real: list_real > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Khovanskii_OKhovanskii_001_062_It__Nat__Onat_Mtf__a_J,type,
khovanskii_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat_a > $o ).
thf(sy_c_Khovanskii_OKhovanskii_001t__Nat__Onat,type,
khovanskii_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii_001tf__a,type,
khovanskii_a: set_a > ( a > a > a ) > a > set_a > $o ).
thf(sy_c_Khovanskii_OKhovanskii_OGmult_001_062_It__Nat__Onat_Mtf__a_J,type,
gmult_nat_a: ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > a ) > nat > nat > a ).
thf(sy_c_Khovanskii_OKhovanskii_OGmult_001t__Nat__Onat,type,
gmult_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Khovanskii_OKhovanskii_OGmult_001tf__a,type,
gmult_a: ( a > a > a ) > a > a > nat > a ).
thf(sy_c_Khovanskii_OKhovanskii_O_092_060alpha_062_001_062_It__Nat__Onat_Mtf__a_J,type,
alpha_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat_a > list_nat > nat > a ).
thf(sy_c_Khovanskii_OKhovanskii_O_092_060alpha_062_001t__Nat__Onat,type,
alpha_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > list_nat > nat ).
thf(sy_c_Khovanskii_OKhovanskii_O_092_060alpha_062_001tf__a,type,
alpha_a: set_a > ( a > a > a ) > a > set_a > list_nat > a ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001_062_It__Nat__Onat_Mtf__a_J,type,
aA_nat_a: set_nat_a > list_nat_a ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001t__Nat__Onat,type,
aA_nat: set_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001tf__a,type,
aA_a: set_a > list_a ).
thf(sy_c_Khovanskii_OKhovanskii_Ouseless_001tf__a,type,
useless_a: set_a > ( a > a > a ) > a > set_a > list_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
inf_inf_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
sup_sup_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
map_Pr3957463622062228029st_nat: ( produc1828647624359046049st_nat > list_nat ) > list_P7940050157051400743st_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_Pr6967709084026004015et_nat: ( produc7819656566062154093et_nat > set_nat ) > list_P6254988961118846195et_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
map_Pr8866779738092938811_set_a: ( produc1703568184450464039_set_a > set_a ) > list_P3660316430366008877_set_a > list_set_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Onth_001_062_It__Nat__Onat_Mtf__a_J,type,
nth_nat_a: list_nat_a > nat > nat > a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_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_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
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_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
zip_li7157463729305086713st_nat: list_list_nat > list_list_nat > list_P7940050157051400743st_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).
thf(sy_c_List_Ozip_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
zip_set_a_set_a: list_set_a > list_set_a > list_P3660316430366008877_set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat_001_062_It__Nat__Onat_Mtf__a_J,type,
rec_nat_nat_a: ( nat > a ) > ( nat > ( nat > a ) > nat > a ) > nat > nat > a ).
thf(sy_c_Nat_Oold_Onat_Orec__nat_001t__Nat__Onat,type,
rec_nat_nat: nat > ( nat > nat > nat ) > nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat_001tf__a,type,
rec_nat_a: a > ( nat > a > a ) > nat > a ).
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__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_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_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_nat > 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_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
bot_bot_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $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__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $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_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_ORuzsa__distance_001tf__a,type,
pluenn5761198478017115492ance_a: set_a > ( a > a > a ) > a > set_a > set_a > real ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001tf__a,type,
pluenn2534204936789923946sset_a: set_a > ( a > a > a ) > a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominussetp_001tf__a,type,
pluenn1126946703085653920setp_a: set_a > ( a > a > a ) > a > ( a > $o ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001tf__a,type,
pluenn1960970773371692859ated_a: set_a > ( a > a > a ) > a > set_a > nat > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc6026655337631754702st_nat: ( list_nat > list_nat > list_nat ) > produc1828647624359046049st_nat > list_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc8983872132230954816et_nat: ( set_nat > set_nat > set_nat ) > produc7819656566062154093et_nat > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc4474087116711199794_set_a: ( set_a > set_a > set_a ) > produc1703568184450464039_set_a > set_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_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_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or6836045993805503595st_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or4185896845444216793st_nat: list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__List__Olist_It__Nat__Onat_J,type,
set_or3033090826390029821st_nat: list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_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_A,type,
a2: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_x,type,
x: list_nat ).
thf(sy_v_y,type,
y: list_nat ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1270)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_left__commute,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B @ g )
=> ( ( member_a @ C @ g )
=> ( ( addition @ B @ ( addition @ A @ C ) )
= ( addition @ A @ ( addition @ B @ C ) ) ) ) ) ) ).
% left_commute
thf(fact_2_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_3_fincomp__unit__eqI,axiom,
! [A2: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= zero ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_unit_eqI
thf(fact_4_fincomp__unit__eqI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( F @ X2 )
= zero ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_unit_eqI
thf(fact_5_fincomp__unit__eqI,axiom,
! [A2: set_nat,F: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= zero ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_unit_eqI
thf(fact_6_Gmult__add__add,axiom,
! [A: a,M: nat,N: nat] :
( ( member_a @ A @ g )
=> ( ( gmult_a @ addition @ zero @ A @ ( plus_plus_nat @ M @ N ) )
= ( addition @ ( gmult_a @ addition @ zero @ A @ M ) @ ( gmult_a @ addition @ zero @ A @ N ) ) ) ) ).
% Gmult_add_add
thf(fact_7_Khovanskii__axioms,axiom,
khovanskii_a @ g @ addition @ zero @ a2 ).
% Khovanskii_axioms
thf(fact_8_associative,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B @ g )
=> ( ( member_a @ C @ g )
=> ( ( addition @ ( addition @ A @ B ) @ C )
= ( addition @ A @ ( addition @ B @ C ) ) ) ) ) ) ).
% associative
thf(fact_9_composition__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B @ g )
=> ( member_a @ ( addition @ A @ B ) @ g ) ) ) ).
% composition_closed
thf(fact_10_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_11_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_12_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_13__092_060alpha_062__def,axiom,
( ( alpha_a @ g @ addition @ zero @ a2 )
= ( ^ [X3: list_nat] :
( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [I: nat] : ( gmult_a @ addition @ zero @ ( nth_a @ ( aA_a @ a2 ) @ I ) @ ( nth_nat @ X3 @ I ) )
@ ( set_ord_lessThan_nat @ ( finite_card_a @ a2 ) ) ) ) ) ).
% \<alpha>_def
thf(fact_14_Gmult__in__G,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ g )
=> ( member_a @ ( gmult_a @ addition @ zero @ A @ N ) @ g ) ) ).
% Gmult_in_G
thf(fact_15_fincomp__unit,axiom,
! [A2: set_nat] :
( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [I: nat] : zero
@ A2 )
= zero ) ).
% fincomp_unit
thf(fact_16_fincomp__unit,axiom,
! [A2: set_nat_a] :
( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [I: nat > a] : zero
@ A2 )
= zero ) ).
% fincomp_unit
thf(fact_17_fincomp__unit,axiom,
! [A2: set_a] :
( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [I: a] : zero
@ A2 )
= zero ) ).
% fincomp_unit
thf(fact_18_M__ify__def,axiom,
! [X: a] :
( ( ( member_a @ X @ g )
=> ( ( commutative_M_ify_a @ g @ zero @ X )
= X ) )
& ( ~ ( member_a @ X @ g )
=> ( ( commutative_M_ify_a @ g @ zero @ X )
= zero ) ) ) ).
% M_ify_def
thf(fact_19_alpha__in__G,axiom,
! [X: list_nat] : ( member_a @ ( alpha_a @ g @ addition @ zero @ a2 @ X ) @ g ) ).
% alpha_in_G
thf(fact_20_assms_I2_J,axiom,
( ( size_size_list_nat @ y )
= ( finite_card_a @ a2 ) ) ).
% assms(2)
thf(fact_21_assms_I1_J,axiom,
( ( size_size_list_nat @ x )
= ( finite_card_a @ a2 ) ) ).
% assms(1)
thf(fact_22_Khovanskii_OaA_Ocong,axiom,
aA_a = aA_a ).
% Khovanskii.aA.cong
thf(fact_23_Khovanskii_OGmult_Ocong,axiom,
gmult_a = gmult_a ).
% Khovanskii.Gmult.cong
thf(fact_24_Khovanskii_O_092_060alpha_062_Ocong,axiom,
alpha_a = alpha_a ).
% Khovanskii.\<alpha>.cong
thf(fact_25_plus__list__def,axiom,
( plus_p2116291331692525561st_nat
= ( ^ [Xs: list_list_nat,Ys: list_list_nat] : ( map_Pr3957463622062228029st_nat @ ( produc6026655337631754702st_nat @ plus_plus_list_nat ) @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) ) ) ).
% plus_list_def
thf(fact_26_plus__list__def,axiom,
( plus_plus_list_nat
= ( ^ [Xs: list_nat,Ys: list_nat] : ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ plus_plus_nat ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% plus_list_def
thf(fact_27_fincomp__const,axiom,
! [A: a,A2: set_nat] :
( ( member_a @ A @ g )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_a @ zero
@ ^ [U2: nat] : ( addition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_28_fincomp__const,axiom,
! [A: a,A2: set_nat_a] :
( ( member_a @ A @ g )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [X3: nat > a] : A
@ A2 )
= ( rec_nat_a @ zero
@ ^ [U2: nat] : ( addition @ A )
@ ( finite_card_nat_a @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_29_fincomp__const,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ g )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_a @ zero
@ ^ [U2: nat] : ( addition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ).
% fincomp_const
thf(fact_30_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_31_sumsetp_Ocases,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( addition @ A3 @ B3 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B2 @ B3 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_32_sumsetp_Osimps,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B2 @ B4 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_33_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B2: a > $o,B: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B2 @ B )
=> ( ( member_a @ B @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ ( addition @ A @ B ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_34_Gmult__in__PiG,axiom,
! [F: nat > nat] :
( member_nat_a
@ ^ [I: nat] : ( gmult_a @ addition @ zero @ ( nth_a @ ( aA_a @ a2 ) @ I ) @ ( F @ I ) )
@ ( pi_nat_a @ ( set_ord_lessThan_nat @ ( finite_card_a @ a2 ) )
@ ^ [Uu: nat] : g ) ) ).
% Gmult_in_PiG
thf(fact_35_nth__aA__in__G,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_a @ a2 ) )
=> ( member_a @ ( nth_a @ ( aA_a @ a2 ) @ I2 ) @ g ) ) ).
% nth_aA_in_G
thf(fact_36_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_37_map2__map__map,axiom,
! [H: nat > nat > nat,F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: product_prod_nat_nat > nat] :
( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) @ ( map_Pr3938374229010428429at_nat @ G @ Xs2 ) ) )
= ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_38_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_39_Gmult__add__diff,axiom,
! [A: a,N: nat,K: nat] :
( ( member_a @ A @ g )
=> ( ( addition @ ( gmult_a @ addition @ zero @ A @ ( plus_plus_nat @ N @ K ) ) @ ( group_inverse_a @ g @ addition @ zero @ ( gmult_a @ addition @ zero @ A @ N ) ) )
= ( gmult_a @ addition @ zero @ A @ K ) ) ) ).
% Gmult_add_diff
thf(fact_40_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_41_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_42_fincomp__cong_H,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_43_fincomp__cong_H,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A2 = B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 )
= ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_44_fincomp__cong_H,axiom,
! [A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_45_fincomp__comp,axiom,
! [F: nat > a,A2: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [X3: nat] : ( addition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( addition @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_46_fincomp__comp,axiom,
! [F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [X3: nat > a] : ( addition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( addition @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 ) @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_47_fincomp__comp,axiom,
! [F: a > a,A2: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [X3: a] : ( addition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( addition @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_48_fincomp__closed,axiom,
! [F: nat > a,F2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : g ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ F2 ) @ g ) ) ).
% fincomp_closed
thf(fact_49_fincomp__closed,axiom,
! [F: ( nat > a ) > a,F2: set_nat_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : g ) )
=> ( member_a @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ F2 ) @ g ) ) ).
% fincomp_closed
thf(fact_50_fincomp__closed,axiom,
! [F: a > a,F2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : g ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ F2 ) @ g ) ) ).
% fincomp_closed
thf(fact_51_length__map,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat] :
( ( size_size_list_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) )
= ( size_s5460976970255530739at_nat @ Xs2 ) ) ).
% length_map
thf(fact_52_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_53_fincomp__inverse,axiom,
! [F: nat > a,A2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [X3: nat] : ( group_inverse_a @ g @ addition @ zero @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ g @ addition @ zero @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 ) ) ) ) ).
% fincomp_inverse
thf(fact_54_fincomp__inverse,axiom,
! [F: ( nat > a ) > a,A2: set_nat_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [X3: nat > a] : ( group_inverse_a @ g @ addition @ zero @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ g @ addition @ zero @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 ) ) ) ) ).
% fincomp_inverse
thf(fact_55_fincomp__inverse,axiom,
! [F: a > a,A2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [X3: a] : ( group_inverse_a @ g @ addition @ zero @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ g @ addition @ zero @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 ) ) ) ) ).
% fincomp_inverse
thf(fact_56_nth__map,axiom,
! [N: nat,Xs2: list_a,F: a > a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( map_a_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_57_nth__map,axiom,
! [N: nat,Xs2: list_a,F: a > nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_nat @ ( map_a_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_58_nth__map,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat,F: product_prod_nat_nat > nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_59_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > a] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_a @ ( map_nat_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_60_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_61_nth__plus__list,axiom,
! [I2: nat,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( nth_list_nat @ ( plus_p2116291331692525561st_nat @ Xs2 @ Ys2 ) @ I2 )
= ( plus_plus_list_nat @ ( nth_list_nat @ Xs2 @ I2 ) @ ( nth_list_nat @ Ys2 @ I2 ) ) ) ) ) ).
% nth_plus_list
thf(fact_62_nth__plus__list,axiom,
! [I2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( plus_plus_list_nat @ Xs2 @ Ys2 ) @ I2 )
= ( plus_plus_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ).
% nth_plus_list
thf(fact_63_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_72_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_73_nth__equalityI,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys2 @ I3 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_74_nth__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_75_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: a] : ( P @ I @ X4 ) ) )
= ( ? [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_a @ Xs @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_76_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X4: nat] : ( P @ I @ X4 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P @ I @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_77_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_78_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_79_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
= ( ^ [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I )
= ( nth_a @ Ys @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_80_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
= ( ^ [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_81_map__eq__imp__length__eq,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: product_prod_nat_nat > nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( map_Pr3938374229010428429at_nat @ F @ Xs2 )
= ( map_Pr3938374229010428429at_nat @ G @ Ys2 ) )
=> ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_82_map__eq__imp__length__eq,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_Pr3938374229010428429at_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_83_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: product_prod_nat_nat > nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Pr3938374229010428429at_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_84_map__equality__iff,axiom,
! [F: a > nat,Xs2: list_a,G: product_prod_nat_nat > nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( map_a_nat @ F @ Xs2 )
= ( map_Pr3938374229010428429at_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_a @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( F @ ( nth_a @ Xs2 @ I ) )
= ( G @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_85_map__equality__iff,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: a > nat,Ys2: list_a] :
( ( ( map_Pr3938374229010428429at_nat @ F @ Xs2 )
= ( map_a_nat @ G @ Ys2 ) )
= ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ys2 ) )
=> ( ( F @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
= ( G @ ( nth_a @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_86_map__equality__iff,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: product_prod_nat_nat > nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( map_Pr3938374229010428429at_nat @ F @ Xs2 )
= ( map_Pr3938374229010428429at_nat @ G @ Ys2 ) )
= ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( F @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
= ( G @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_87_map__equality__iff,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_Pr3938374229010428429at_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
= ( G @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_88_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: product_prod_nat_nat > nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Pr3938374229010428429at_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I ) )
= ( G @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_89_Khovanskii_OGmult__add__diff,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a,N: nat,K: nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( ( Addition @ ( gmult_nat_a @ Addition @ Zero @ A @ ( plus_plus_nat @ N @ K ) ) @ ( group_inverse_nat_a @ G2 @ Addition @ Zero @ ( gmult_nat_a @ Addition @ Zero @ A @ N ) ) )
= ( gmult_nat_a @ Addition @ Zero @ A @ K ) ) ) ) ).
% Khovanskii.Gmult_add_diff
thf(fact_90_Khovanskii_OGmult__add__diff,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,N: nat,K: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ G2 )
=> ( ( Addition @ ( gmult_nat @ Addition @ Zero @ A @ ( plus_plus_nat @ N @ K ) ) @ ( group_inverse_nat @ G2 @ Addition @ Zero @ ( gmult_nat @ Addition @ Zero @ A @ N ) ) )
= ( gmult_nat @ Addition @ Zero @ A @ K ) ) ) ) ).
% Khovanskii.Gmult_add_diff
thf(fact_91_Khovanskii_OGmult__add__diff,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,N: nat,K: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ G2 )
=> ( ( Addition @ ( gmult_a @ Addition @ Zero @ A @ ( plus_plus_nat @ N @ K ) ) @ ( group_inverse_a @ G2 @ Addition @ Zero @ ( gmult_a @ Addition @ Zero @ A @ N ) ) )
= ( gmult_a @ Addition @ Zero @ A @ K ) ) ) ) ).
% Khovanskii.Gmult_add_diff
thf(fact_92_Khovanskii_OGmult__in__G,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a,N: nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( member_nat_a @ ( gmult_nat_a @ Addition @ Zero @ A @ N ) @ G2 ) ) ) ).
% Khovanskii.Gmult_in_G
thf(fact_93_Khovanskii_OGmult__in__G,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,N: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ G2 )
=> ( member_nat @ ( gmult_nat @ Addition @ Zero @ A @ N ) @ G2 ) ) ) ).
% Khovanskii.Gmult_in_G
thf(fact_94_Khovanskii_OGmult__in__G,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,N: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ G2 )
=> ( member_a @ ( gmult_a @ Addition @ Zero @ A @ N ) @ G2 ) ) ) ).
% Khovanskii.Gmult_in_G
thf(fact_95_Khovanskii_Onth__aA__in__G,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,I2: nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( ord_less_nat @ I2 @ ( finite_card_nat_a @ A2 ) )
=> ( member_nat_a @ ( nth_nat_a @ ( aA_nat_a @ A2 ) @ I2 ) @ G2 ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_96_Khovanskii_Onth__aA__in__G,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,I2: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( ord_less_nat @ I2 @ ( finite_card_nat @ A2 ) )
=> ( member_nat @ ( nth_nat @ ( aA_nat @ A2 ) @ I2 ) @ G2 ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_97_Khovanskii_Onth__aA__in__G,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,I2: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( ord_less_nat @ I2 @ ( finite_card_a @ A2 ) )
=> ( member_a @ ( nth_a @ ( aA_a @ A2 ) @ I2 ) @ G2 ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_98_Khovanskii_Oalpha__in__G,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,X: list_nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( member_nat_a @ ( alpha_nat_a @ G2 @ Addition @ Zero @ A2 @ X ) @ G2 ) ) ).
% Khovanskii.alpha_in_G
thf(fact_99_Khovanskii_Oalpha__in__G,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X: list_nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( member_nat @ ( alpha_nat @ G2 @ Addition @ Zero @ A2 @ X ) @ G2 ) ) ).
% Khovanskii.alpha_in_G
thf(fact_100_Khovanskii_Oalpha__in__G,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: list_nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( member_a @ ( alpha_a @ G2 @ Addition @ Zero @ A2 @ X ) @ G2 ) ) ).
% Khovanskii.alpha_in_G
thf(fact_101_Khovanskii_OGmult__add__add,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a,M: nat,N: nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( ( gmult_nat_a @ Addition @ Zero @ A @ ( plus_plus_nat @ M @ N ) )
= ( Addition @ ( gmult_nat_a @ Addition @ Zero @ A @ M ) @ ( gmult_nat_a @ Addition @ Zero @ A @ N ) ) ) ) ) ).
% Khovanskii.Gmult_add_add
thf(fact_102_Khovanskii_OGmult__add__add,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,M: nat,N: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ G2 )
=> ( ( gmult_nat @ Addition @ Zero @ A @ ( plus_plus_nat @ M @ N ) )
= ( Addition @ ( gmult_nat @ Addition @ Zero @ A @ M ) @ ( gmult_nat @ Addition @ Zero @ A @ N ) ) ) ) ) ).
% Khovanskii.Gmult_add_add
thf(fact_103_Khovanskii_OGmult__add__add,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,M: nat,N: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ G2 )
=> ( ( gmult_a @ Addition @ Zero @ A @ ( plus_plus_nat @ M @ N ) )
= ( Addition @ ( gmult_a @ Addition @ Zero @ A @ M ) @ ( gmult_a @ Addition @ Zero @ A @ N ) ) ) ) ) ).
% Khovanskii.Gmult_add_add
thf(fact_104_Khovanskii_OGmult__in__PiG,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,F: nat > nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( member_nat_nat
@ ^ [I: nat] : ( gmult_nat @ Addition @ Zero @ ( nth_nat @ ( aA_nat @ A2 ) @ I ) @ ( F @ I ) )
@ ( pi_nat_nat @ ( set_ord_lessThan_nat @ ( finite_card_nat @ A2 ) )
@ ^ [Uu: nat] : G2 ) ) ) ).
% Khovanskii.Gmult_in_PiG
thf(fact_105_Khovanskii_OGmult__in__PiG,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,F: nat > nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( member_nat_a
@ ^ [I: nat] : ( gmult_a @ Addition @ Zero @ ( nth_a @ ( aA_a @ A2 ) @ I ) @ ( F @ I ) )
@ ( pi_nat_a @ ( set_ord_lessThan_nat @ ( finite_card_a @ A2 ) )
@ ^ [Uu: nat] : G2 ) ) ) ).
% Khovanskii.Gmult_in_PiG
thf(fact_106_Khovanskii_O_092_060alpha_062__def,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( alpha_nat @ G2 @ Addition @ Zero @ A2 )
= ( ^ [X3: list_nat] :
( commut1028764413824576968at_nat @ G2 @ Addition @ Zero
@ ^ [I: nat] : ( gmult_nat @ Addition @ Zero @ ( nth_nat @ ( aA_nat @ A2 ) @ I ) @ ( nth_nat @ X3 @ I ) )
@ ( set_ord_lessThan_nat @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).
% Khovanskii.\<alpha>_def
thf(fact_107_Khovanskii_O_092_060alpha_062__def,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( alpha_a @ G2 @ Addition @ Zero @ A2 )
= ( ^ [X3: list_nat] :
( commut6741328216151336360_a_nat @ G2 @ Addition @ Zero
@ ^ [I: nat] : ( gmult_a @ Addition @ Zero @ ( nth_a @ ( aA_a @ A2 ) @ I ) @ ( nth_nat @ X3 @ I ) )
@ ( set_ord_lessThan_nat @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% Khovanskii.\<alpha>_def
thf(fact_108_minussetp_Osimps,axiom,
! [A2: a > $o,A: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ A )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g ) ) ) ) ).
% minussetp.simps
thf(fact_109_minussetp_OminussetI,axiom,
! [A2: a > $o,A: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ ( group_inverse_a @ g @ addition @ zero @ A ) ) ) ) ).
% minussetp.minussetI
thf(fact_110_minussetp_Ocases,axiom,
! [A2: a > $o,A: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ A )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
=> ( ( A2 @ A3 )
=> ~ ( member_a @ A3 @ g ) ) ) ) ).
% minussetp.cases
thf(fact_111_comp__fun__commute__onI,axiom,
! [F: nat > a,F2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : g ) )
=> ( finite1071566134745755356_nat_a @ F2
@ ^ [X3: nat,Y3: a] : ( addition @ ( F @ X3 ) @ ( commutative_M_ify_a @ g @ zero @ Y3 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_112_idx__less__cardA,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ a2 @ A ) @ ( finite_card_a @ a2 ) ) ) ).
% idx_less_cardA
thf(fact_113_power__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ g )
=> ( ( member_a @ A @ g )
=> ( ( rec_nat_a @ zero
@ ^ [U2: nat] : ( addition @ A )
@ ( finite_card_a @ g ) )
= zero ) ) ) ).
% power_order_eq_one
thf(fact_114_fincomp__Suc3,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ N ) )
= ( addition @ ( F @ N ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% fincomp_Suc3
thf(fact_115_fincomp__singleton__swap,axiom,
! [I2: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [J: nat] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton_swap
thf(fact_116_fincomp__singleton__swap,axiom,
! [I2: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [J: nat > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton_swap
thf(fact_117_fincomp__singleton__swap,axiom,
! [I2: a,A2: set_a,F: a > a] :
( ( member_a @ I2 @ A2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [J: a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton_swap
thf(fact_118_fincomp__singleton,axiom,
! [I2: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [J: nat] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton
thf(fact_119_fincomp__singleton,axiom,
! [I2: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero
@ ^ [J: nat > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton
thf(fact_120_fincomp__singleton,axiom,
! [I2: a,A2: set_a,F: a > a] :
( ( member_a @ I2 @ A2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero
@ ^ [J: a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ zero )
@ A2 )
= ( F @ I2 ) ) ) ) ) ).
% fincomp_singleton
thf(fact_121_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_122_finA,axiom,
finite_finite_a @ a2 ).
% finA
thf(fact_123_fincomp__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_infinite
thf(fact_124_fincomp__infinite,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_infinite
thf(fact_125_fincomp__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ).
% fincomp_infinite
thf(fact_126_aA__idx__eq,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ( nth_a @ ( aA_a @ a2 ) @ ( counta3566351752493190365t_on_a @ a2 @ A ) )
= A ) ) ).
% aA_idx_eq
thf(fact_127_Khovanskii_OfinA,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( finite_finite_nat @ A2 ) ) ).
% Khovanskii.finA
thf(fact_128_Khovanskii_OfinA,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( finite_finite_a @ A2 ) ) ).
% Khovanskii.finA
thf(fact_129_finite__maxlen,axiom,
! [M2: set_list_nat] :
( ( finite8100373058378681591st_nat @ M2 )
=> ? [N2: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M2 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_130_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_131_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_132_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_133_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_134_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_135_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_136_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_137_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_138_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_139_Khovanskii_Oidx__less__cardA,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ A2 )
=> ( ord_less_nat @ ( counta7388732360187139652_nat_a @ A2 @ A ) @ ( finite_card_nat_a @ A2 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_140_Khovanskii_Oidx__less__cardA,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_nat @ ( counta4844910239362777137on_nat @ A2 @ A ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_141_Khovanskii_Oidx__less__cardA,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ A2 @ A ) @ ( finite_card_a @ A2 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_142_Khovanskii_OaA__idx__eq,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ A2 )
=> ( ( nth_nat_a @ ( aA_nat_a @ A2 ) @ ( counta7388732360187139652_nat_a @ A2 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_143_Khovanskii_OaA__idx__eq,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( nth_nat @ ( aA_nat @ A2 ) @ ( counta4844910239362777137on_nat @ A2 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_144_Khovanskii_OaA__idx__eq,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( nth_a @ ( aA_a @ A2 ) @ ( counta3566351752493190365t_on_a @ A2 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_145_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_146_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_147_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_148_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_149_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_150_trans__less__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_151_trans__less__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_152_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_153_fincomp__def,axiom,
! [A2: set_nat,F: nat > a] :
( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y3: a] : ( addition @ ( F @ X3 ) @ ( commutative_M_ify_a @ g @ zero @ Y3 ) )
@ zero
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ A2 )
= zero ) ) ) ).
% fincomp_def
thf(fact_154_fincomp__def,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ( ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 )
= ( finite_fold_nat_a_a
@ ^ [X3: nat > a,Y3: a] : ( addition @ ( F @ X3 ) @ ( commutative_M_ify_a @ g @ zero @ Y3 ) )
@ zero
@ A2 ) ) )
& ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ) ).
% fincomp_def
thf(fact_155_fincomp__def,axiom,
! [A2: set_a,F: a > a] :
( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y3: a] : ( addition @ ( F @ X3 ) @ ( commutative_M_ify_a @ g @ zero @ Y3 ) )
@ zero
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ A2 )
= zero ) ) ) ).
% fincomp_def
thf(fact_156_commutative__monoid_Ofincomp__Suc3,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ N ) )
= ( Composition @ ( F @ N ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Suc3
thf(fact_157_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,F2: set_nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M2 ) )
=> ( finite1071566134745755356_nat_a @ F2
@ ^ [X3: nat,Y3: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M2 @ Unit @ Y3 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_158_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,A2: set_a] :
( ( group_3093379471365697572_nat_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M2 )
=> ( ( commut1274061894236046463at_a_a @ M2 @ Composition @ Unit
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_159_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,A2: set_a] :
( ( group_6791354081887936081id_nat @ M2 @ Composition @ Unit )
=> ( ( member_nat @ A @ M2 )
=> ( ( commut1549887680474846982_nat_a @ M2 @ Composition @ Unit
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_nat @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_160_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,A2: set_nat] :
( ( group_3093379471365697572_nat_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M2 )
=> ( ( commut6753747983606973455_a_nat @ M2 @ Composition @ Unit
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_161_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,A2: set_nat] :
( ( group_6791354081887936081id_nat @ M2 @ Composition @ Unit )
=> ( ( member_nat @ A @ M2 )
=> ( ( commut1028764413824576968at_nat @ M2 @ Composition @ Unit
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_nat @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_162_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ A @ M2 )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [X3: nat] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_163_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ A @ M2 )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit
@ ^ [X3: nat > a] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_164_commutative__monoid_Ofincomp__const,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A: a,A2: set_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ A @ M2 )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit
@ ^ [X3: a] : A
@ A2 )
= ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_a @ A2 ) ) ) ) ) ).
% commutative_monoid.fincomp_const
thf(fact_165_abelian__group_Ofincomp__inverse,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : G2 ) )
=> ( ( commut6741328216151336360_a_nat @ G2 @ Composition @ Unit
@ ^ [X3: nat] : ( group_inverse_a @ G2 @ Composition @ Unit @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ G2 @ Composition @ Unit @ ( commut6741328216151336360_a_nat @ G2 @ Composition @ Unit @ F @ A2 ) ) ) ) ) ).
% abelian_group.fincomp_inverse
thf(fact_166_abelian__group_Ofincomp__inverse,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,A2: set_nat_a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : G2 ) )
=> ( ( commut5242989786243415821_nat_a @ G2 @ Composition @ Unit
@ ^ [X3: nat > a] : ( group_inverse_a @ G2 @ Composition @ Unit @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ G2 @ Composition @ Unit @ ( commut5242989786243415821_nat_a @ G2 @ Composition @ Unit @ F @ A2 ) ) ) ) ) ).
% abelian_group.fincomp_inverse
thf(fact_167_abelian__group_Ofincomp__inverse,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,F: a > a,A2: set_a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : G2 ) )
=> ( ( commut5005951359559292710mp_a_a @ G2 @ Composition @ Unit
@ ^ [X3: a] : ( group_inverse_a @ G2 @ Composition @ Unit @ ( F @ X3 ) )
@ A2 )
= ( group_inverse_a @ G2 @ Composition @ Unit @ ( commut5005951359559292710mp_a_a @ G2 @ Composition @ Unit @ F @ A2 ) ) ) ) ) ).
% abelian_group.fincomp_inverse
thf(fact_168_abelian__group_Opower__order__eq__one,axiom,
! [G2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a] :
( ( group_2359011037596659067_nat_a @ G2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ G2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( ( rec_nat_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat_a @ G2 ) )
= Unit ) ) ) ) ).
% abelian_group.power_order_eq_one
thf(fact_169_abelian__group_Opower__order__eq__one,axiom,
! [G2: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat] :
( ( group_4000446350026676922up_nat @ G2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ G2 )
=> ( ( member_nat @ A @ G2 )
=> ( ( rec_nat_nat @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_nat @ G2 ) )
= Unit ) ) ) ) ).
% abelian_group.power_order_eq_one
thf(fact_170_abelian__group_Opower__order__eq__one,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( ( finite_finite_a @ G2 )
=> ( ( member_a @ A @ G2 )
=> ( ( rec_nat_a @ Unit
@ ^ [U2: nat] : ( Composition @ A )
@ ( finite_card_a @ G2 ) )
= Unit ) ) ) ) ).
% abelian_group.power_order_eq_one
thf(fact_171_lessThan__iff,axiom,
! [I2: list_nat,K: list_nat] :
( ( member_list_nat @ I2 @ ( set_or3033090826390029821st_nat @ K ) )
= ( ord_less_list_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_172_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_173_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_174_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_175_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_176_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_177_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_178_commutative__monoid_Ofincomp__singleton__swap,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: nat,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat @ I2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [J: nat] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton_swap
thf(fact_179_commutative__monoid_Ofincomp__singleton__swap,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ I2 @ A2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit
@ ^ [J: nat > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton_swap
thf(fact_180_commutative__monoid_Ofincomp__singleton__swap,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ I2 @ A2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit
@ ^ [J: a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton_swap
thf(fact_181_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_182_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_183_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_184_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_185_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_186_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_187_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_188_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_189_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_190_add_Oleft__commute,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ B @ ( plus_plus_list_nat @ A @ C ) )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_191_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_192_add_Ocommute,axiom,
( plus_plus_list_nat
= ( ^ [A4: list_nat,B4: list_nat] : ( plus_plus_list_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_193_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_194_add_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A @ B ) @ C )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_195_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_196_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_197_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_198_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A @ B ) @ C )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_200_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_201_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_202_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_203_commutative__monoid_Ofincomp__def,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y3: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M2 @ Unit @ Y3 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_204_commutative__monoid_Ofincomp__def,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 )
= ( finite_fold_nat_a_a
@ ^ [X3: nat > a,Y3: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M2 @ Unit @ Y3 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_205_commutative__monoid_Ofincomp__def,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y3: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M2 @ Unit @ Y3 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_206_commutative__monoid_Ofincomp_Ocong,axiom,
commut6741328216151336360_a_nat = commut6741328216151336360_a_nat ).
% commutative_monoid.fincomp.cong
thf(fact_207_commutative__monoid_Ofincomp_Ocong,axiom,
commut5242989786243415821_nat_a = commut5242989786243415821_nat_a ).
% commutative_monoid.fincomp.cong
thf(fact_208_commutative__monoid_Ofincomp_Ocong,axiom,
commut5005951359559292710mp_a_a = commut5005951359559292710mp_a_a ).
% commutative_monoid.fincomp.cong
thf(fact_209_commutative__monoid_Oleft__commute,axiom,
! [M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,B: nat > a,C: nat > a] :
( ( group_3093379471365697572_nat_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M2 )
=> ( ( member_nat_a @ B @ M2 )
=> ( ( member_nat_a @ C @ M2 )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_210_commutative__monoid_Oleft__commute,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat,C: nat] :
( ( group_6791354081887936081id_nat @ M2 @ Composition @ Unit )
=> ( ( member_nat @ A @ M2 )
=> ( ( member_nat @ B @ M2 )
=> ( ( member_nat @ C @ M2 )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_211_commutative__monoid_Oleft__commute,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A: a,B: a,C: a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ A @ M2 )
=> ( ( member_a @ B @ M2 )
=> ( ( member_a @ C @ M2 )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_212_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_213_commutative__monoid_OM__ify_Ocong,axiom,
commutative_M_ify_a = commutative_M_ify_a ).
% commutative_monoid.M_ify.cong
thf(fact_214_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_215_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_216_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_217_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_218_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_219_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_220_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_221_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_222_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_223_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_224_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_225_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_226_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_227_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_228_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_229_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_230_lessThan__strict__subset__iff,axiom,
! [M: list_nat,N: list_nat] :
( ( ord_le1190675801316882794st_nat @ ( set_or3033090826390029821st_nat @ M ) @ ( set_or3033090826390029821st_nat @ N ) )
= ( ord_less_list_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_231_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_232_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_233_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_234_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_235_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_236_lessThan__def,axiom,
( set_or3033090826390029821st_nat
= ( ^ [U2: list_nat] :
( collect_list_nat
@ ^ [X3: list_nat] : ( ord_less_list_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_237_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_238_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_239_commutative__monoid_OM__ify__def,axiom,
! [M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a] :
( ( group_3093379471365697572_nat_a @ M2 @ Composition @ Unit )
=> ( ( ( member_nat_a @ X @ M2 )
=> ( ( commut2316704705022288065_nat_a @ M2 @ Unit @ X )
= X ) )
& ( ~ ( member_nat_a @ X @ M2 )
=> ( ( commut2316704705022288065_nat_a @ M2 @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_240_commutative__monoid_OM__ify__def,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_6791354081887936081id_nat @ M2 @ Composition @ Unit )
=> ( ( ( member_nat @ X @ M2 )
=> ( ( commut810702690453168372fy_nat @ M2 @ Unit @ X )
= X ) )
& ( ~ ( member_nat @ X @ M2 )
=> ( ( commut810702690453168372fy_nat @ M2 @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_241_commutative__monoid_OM__ify__def,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( ( member_a @ X @ M2 )
=> ( ( commutative_M_ify_a @ M2 @ Unit @ X )
= X ) )
& ( ~ ( member_a @ X @ M2 )
=> ( ( commutative_M_ify_a @ M2 @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_242_commutative__monoid_Ofincomp__unit,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [I: nat] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_243_commutative__monoid_Ofincomp__unit,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit
@ ^ [I: nat > a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_244_commutative__monoid_Ofincomp__unit,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit
@ ^ [I: a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_245_commutative__monoid_Ofincomp__infinite,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_246_commutative__monoid_Ofincomp__infinite,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_247_commutative__monoid_Ofincomp__infinite,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_248_commutative__monoid_Ofincomp__closed,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,F2: set_nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M2 ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ F2 ) @ M2 ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_249_commutative__monoid_Ofincomp__closed,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,F2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : M2 ) )
=> ( member_a @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ F2 ) @ M2 ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_250_commutative__monoid_Ofincomp__closed,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: a > a,F2: set_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M2 ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ F2 ) @ M2 ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_251_commutative__monoid_Ofincomp__cong_H,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 )
= ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_252_commutative__monoid_Ofincomp__cong_H,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 )
= ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_253_commutative__monoid_Ofincomp__cong_H,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ B2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_254_commutative__monoid_Ofincomp__comp,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [X3: nat] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_255_commutative__monoid_Ofincomp__comp,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit
@ ^ [X3: nat > a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ A2 ) @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_256_commutative__monoid_Ofincomp__comp,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: a > a,A2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit
@ ^ [X3: a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_257_commutative__monoid_Ofincomp__singleton,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: nat,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat @ I2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [J: nat] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton
thf(fact_258_commutative__monoid_Ofincomp__singleton,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ I2 @ A2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit
@ ^ [J: nat > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton
thf(fact_259_commutative__monoid_Ofincomp__singleton,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,I2: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ I2 @ A2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit
@ ^ [J: a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ Unit )
@ A2 )
= ( F @ I2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_singleton
thf(fact_260_finite__sumset__iterated,axiom,
! [A2: set_a,R: nat] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R ) ) ) ).
% finite_sumset_iterated
thf(fact_261_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_262_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_263_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_264_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_265_minussetp__minusset__eq,axiom,
! [A2: set_a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ) ).
% minussetp_minusset_eq
thf(fact_266_minusset__def,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero )
= ( ^ [A5: set_a] :
( collect_a
@ ( pluenn1126946703085653920setp_a @ g @ addition @ zero
@ ^ [X3: a] : ( member_a @ X3 @ A5 ) ) ) ) ) ).
% minusset_def
thf(fact_267_abelian__group_Ointro,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( group_4866109990395492029noid_a @ G2 @ Composition @ Unit )
=> ( group_201663378560352916roup_a @ G2 @ Composition @ Unit ) ) ) ).
% abelian_group.intro
thf(fact_268_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_269_local_Oinvertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ g )
& ( ( addition @ U @ X3 )
= zero )
& ( ( addition @ X3 @ U )
= zero ) ) ) ) ) ).
% local.invertible_def
thf(fact_270_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_271_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_272_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_273_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_274_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_275_finite__minusset,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ).
% finite_minusset
thf(fact_276_minusset_Ocases,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ~ ( member_a @ A3 @ g ) ) ) ) ).
% minusset.cases
thf(fact_277_minusset_OminussetI,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ A ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ) ).
% minusset.minussetI
thf(fact_278_minusset_Osimps,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g ) ) ) ) ).
% minusset.simps
thf(fact_279_minusset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% minusset_iterated_minusset
thf(fact_280_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_281_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_282_Units__def,axiom,
( ( group_Units_a @ g @ addition @ zero )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ g )
& ( group_invertible_a @ g @ addition @ zero @ U2 ) ) ) ) ).
% Units_def
thf(fact_283_card__sumset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_284_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_nat @ N5 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_285_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_286_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_287_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_288_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_289_invertible__left__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_290_invertible__right__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_291_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_292_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_293_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_294_local_Oinvertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% local.invertible_left_inverse
thf(fact_295_local_Oinvertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% local.invertible_right_inverse
thf(fact_296_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_297_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_298_group_Oinvertible,axiom,
! [G2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_group_nat_a @ G2 @ Composition @ Unit )
=> ( ( member_nat_a @ U @ G2 )
=> ( group_645299334525884886_nat_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_299_group_Oinvertible,axiom,
! [G2: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_group_nat @ G2 @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( group_invertible_nat @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_300_group_Oinvertible,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( group_invertible_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_301_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_302_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_303_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_304_fold__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z2: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_a_a @ F @ Z2 @ A2 )
= ( finite_fold_a_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_305_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat_a,F: a > ( nat > a ) > nat > a,G: a > ( nat > a ) > nat > a,Z2: nat > a] :
( ! [A3: a,B3: nat > a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: nat > a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( member_nat_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat_a @ Z2 @ B2 )
=> ( ( finite_fold_a_nat_a @ F @ Z2 @ A2 )
= ( finite_fold_a_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_306_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z2: nat] :
( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: a,B3: nat] :
( ( member_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite_fold_a_nat @ F @ Z2 @ A2 )
= ( finite_fold_a_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_307_fold__closed__eq,axiom,
! [A2: set_nat_a,B2: set_a,F: ( nat > a ) > a > a,G: ( nat > a ) > a > a,Z2: a] :
( ! [A3: nat > a,B3: a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat > a,B3: a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_nat_a_a @ F @ Z2 @ A2 )
= ( finite_fold_nat_a_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_308_fold__closed__eq,axiom,
! [A2: set_nat_a,B2: set_nat_a,F: ( nat > a ) > ( nat > a ) > nat > a,G: ( nat > a ) > ( nat > a ) > nat > a,Z2: nat > a] :
( ! [A3: nat > a,B3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat > a,B3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( member_nat_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat_a @ Z2 @ B2 )
=> ( ( finite8825492827046180360_nat_a @ F @ Z2 @ A2 )
= ( finite8825492827046180360_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_309_fold__closed__eq,axiom,
! [A2: set_nat_a,B2: set_nat,F: ( nat > a ) > nat > nat,G: ( nat > a ) > nat > nat,Z2: nat] :
( ! [A3: nat > a,B3: nat] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat > a,B3: nat] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite7774500027257897325_a_nat @ F @ Z2 @ A2 )
= ( finite7774500027257897325_a_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_310_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z2: a] :
( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( member_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_nat_a @ F @ Z2 @ A2 )
= ( finite_fold_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_311_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat_a,F: nat > ( nat > a ) > nat > a,G: nat > ( nat > a ) > nat > a,Z2: nat > a] :
( ! [A3: nat,B3: nat > a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat > a] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( member_nat_a @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat_a @ Z2 @ B2 )
=> ( ( finite6730669110406474827_nat_a @ F @ Z2 @ A2 )
= ( finite6730669110406474827_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_312_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z2: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite_fold_nat_nat @ F @ Z2 @ A2 )
= ( finite_fold_nat_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_313_commutative__monoid_Ocommutative,axiom,
! [M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a] :
( ( group_3093379471365697572_nat_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ X @ M2 )
=> ( ( member_nat_a @ Y @ M2 )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_314_commutative__monoid_Ocommutative,axiom,
! [M2: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_6791354081887936081id_nat @ M2 @ Composition @ Unit )
=> ( ( member_nat @ X @ M2 )
=> ( ( member_nat @ Y @ M2 )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_315_commutative__monoid_Ocommutative,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_a @ X @ M2 )
=> ( ( member_a @ Y @ M2 )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_316_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_a,B2: set_a,R2: ( nat > a ) > a > $o] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A4: nat > a] :
( ( member_nat_a @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_317_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_a,B2: set_nat,R2: ( nat > a ) > nat > $o] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A4: nat > a] :
( ( member_nat_a @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_318_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_a,R2: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_319_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_320_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_a,R2: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_321_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R2 @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_322_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_323_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_324_abelian__group_Oaxioms_I2_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( group_4866109990395492029noid_a @ G2 @ Composition @ Unit ) ) ).
% abelian_group.axioms(2)
thf(fact_325_abelian__group_Oaxioms_I1_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ).
% abelian_group.axioms(1)
thf(fact_326_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_327_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_328_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G3: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G3 @ Composition2 @ Unit2 )
& ( group_4866109990395492029noid_a @ G3 @ Composition2 @ Unit2 ) ) ) ) ).
% abelian_group_def
thf(fact_329_card__minusset_H,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ).
% card_minusset'
thf(fact_330_sumset__iterated__subset__carrier,axiom,
! [A2: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_331_minusset__subset__carrier,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ g ) ).
% minusset_subset_carrier
thf(fact_332_finite__differenceset,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ) ) ).
% finite_differenceset
thf(fact_333_card__differenceset__commute,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ) ).
% card_differenceset_commute
thf(fact_334_minusset__distrib__sum,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ).
% minusset_distrib_sum
thf(fact_335_fincomp__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( addition @ ( F @ ( suc @ N ) ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% fincomp_Suc
thf(fact_336_useless__def,axiom,
! [X: list_nat] :
( ( useless_a @ g @ addition @ zero @ a2 @ X )
= ( ? [Y3: list_nat] :
( ( ord_less_list_nat @ Y3 @ X )
& ( ( groups4561878855575611511st_nat @ Y3 )
= ( groups4561878855575611511st_nat @ X ) )
& ( ( alpha_a @ g @ addition @ zero @ a2 @ Y3 )
= ( alpha_a @ g @ addition @ zero @ a2 @ X ) )
& ( ( size_size_list_nat @ Y3 )
= ( size_size_list_nat @ X ) ) ) ) ) ).
% useless_def
thf(fact_337_sumset__commute,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 ) ) ).
% sumset_commute
thf(fact_338_sumset__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ C2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C2 ) ) ) ).
% sumset_assoc
thf(fact_339_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B @ B2 )
=> ( ( member_a @ B @ g )
=> ( member_a @ ( addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_340_sumset_Osimps,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B4 @ B2 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumset.simps
thf(fact_341_sumset_Ocases,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( addition @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B3 @ B2 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_342_AsubG,axiom,
ord_less_eq_set_a @ a2 @ g ).
% AsubG
thf(fact_343_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_344_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_345_finite__sumset,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset
thf(fact_346_sumset__subset__carrier,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g ) ).
% sumset_subset_carrier
thf(fact_347_sumset__mono,axiom,
! [A7: set_a,A2: set_a,B6: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ( ( ord_less_eq_set_a @ B6 @ B2 )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ B6 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_mono
thf(fact_348_sumsetp__sumset__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A2 )
@ ^ [X3: a] : ( member_a @ X3 @ B2 ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_349_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A5: set_a,B7: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B7 ) ) ) ) ) ).
% sumset_def
thf(fact_350_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_351_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_352_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_353_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_354_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_355_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_356_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_357_atMost__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_358_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_359_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_360_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_361_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_362_old_Onat_Osimps_I7_J,axiom,
! [F1: a,F22: nat > a > a,Nat: nat] :
( ( rec_nat_a @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat_a @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_363_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B7: set_a] : ( ord_less_eq_set_a @ B7 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_364_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B7: set_nat] : ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_365_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_366_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_367_atMost__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_368_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_369_atMost__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_370_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_371_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_372_Gmult__Suc,axiom,
! [A: a,N: nat] :
( ( gmult_a @ addition @ zero @ A @ ( suc @ N ) )
= ( addition @ A @ ( gmult_a @ addition @ zero @ A @ N ) ) ) ).
% Gmult_Suc
thf(fact_373_differenceset__commute,axiom,
! [B2: set_a,A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ).
% differenceset_commute
thf(fact_374_sumset__iterated__Suc,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( suc @ K ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% sumset_iterated_Suc
thf(fact_375_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_376_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_377_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_a @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_378_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_379_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_380_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_real @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_381_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_382_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_383_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_384_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_385_Khovanskii_Ouseless_Ocong,axiom,
useless_a = useless_a ).
% Khovanskii.useless.cong
thf(fact_386_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_387_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_388_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_389_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_390_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_391_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_392_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_393_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_394_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_395_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_396_add__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 @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_397_add__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 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_398_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_399_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_400_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_401_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_402_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_403_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_404_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_405_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_406_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_407_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_408_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_409_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_410_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_411_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_412_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_413_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_414_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_415_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_416_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_417_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_418_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_419_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_420_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_421_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_422_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_423_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_424_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ A @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_425_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_426_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_427_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_428_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ X2 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_429_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_430_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_431_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_432_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_433_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_434_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_435_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_436_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_437_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_438_Khovanskii_OAsubG,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ord_less_eq_set_a @ A2 @ G2 ) ) ).
% Khovanskii.AsubG
thf(fact_439_Khovanskii_OAsubG,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ G2 ) ) ).
% Khovanskii.AsubG
thf(fact_440_atMost__def,axiom,
( set_ord_atMost_set_a
= ( ^ [U2: set_a] :
( collect_set_a
@ ^ [X3: set_a] : ( ord_less_eq_set_a @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_441_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_442_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_443_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_444_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_445_lift__Suc__mono__less,axiom,
! [F: nat > list_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_list_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_list_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_446_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_447_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_448_lift__Suc__mono__less__iff,axiom,
! [F: nat > list_nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_list_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_list_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_449_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_450_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_451_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_452_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_453_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_454_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_455_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_456_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_457_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_458_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_459_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_460_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_461_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
? [K2: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_462_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_463_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B8: set_a] :
( ( finite_finite_a @ B8 )
& ( ( finite_card_a @ B8 )
= N )
& ( ord_less_eq_set_a @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_464_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_465_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_466_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_467_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_468_Khovanskii_OGmult__Suc,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,N: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( gmult_a @ Addition @ Zero @ A @ ( suc @ N ) )
= ( Addition @ A @ ( gmult_a @ Addition @ Zero @ A @ N ) ) ) ) ).
% Khovanskii.Gmult_Suc
thf(fact_469_Khovanskii_Ouseless__def,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: list_nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( useless_a @ G2 @ Addition @ Zero @ A2 @ X )
= ( ? [Y3: list_nat] :
( ( ord_less_list_nat @ Y3 @ X )
& ( ( groups4561878855575611511st_nat @ Y3 )
= ( groups4561878855575611511st_nat @ X ) )
& ( ( alpha_a @ G2 @ Addition @ Zero @ A2 @ Y3 )
= ( alpha_a @ G2 @ Addition @ Zero @ A2 @ X ) )
& ( ( size_size_list_nat @ Y3 )
= ( size_size_list_nat @ X ) ) ) ) ) ) ).
% Khovanskii.useless_def
thf(fact_470_Iic__subset__Iio__iff,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_or4185896845444216793st_nat @ A ) @ ( set_or3033090826390029821st_nat @ B ) )
= ( ord_less_list_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_471_Iic__subset__Iio__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_472_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_473_sum__list__plus,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_plus
thf(fact_474_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_475_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_476_commutative__monoid_Ofincomp__Suc,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( Composition @ ( F @ ( suc @ N ) ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Suc
thf(fact_477_fincomp__Suc2,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( addition
@ ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% fincomp_Suc2
thf(fact_478_subgroupI,axiom,
! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ g )
=> ( ( member_a @ zero @ G2 )
=> ( ! [G4: a,H2: a] :
( ( member_a @ G4 @ G2 )
=> ( ( member_a @ H2 @ G2 )
=> ( member_a @ ( addition @ G4 @ H2 ) @ G2 ) ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G2 )
=> ( group_invertible_a @ g @ addition @ zero @ G4 ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G2 )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G4 ) @ G2 ) )
=> ( group_subgroup_a @ G2 @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_479_fincomp__mono__neutral__cong__right,axiom,
! [B2: set_nat,A2: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ B2 )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero @ H @ A2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_right
thf(fact_480_fincomp__mono__neutral__cong__right,axiom,
! [B2: set_nat_a,A2: set_nat_a,G: ( nat > a ) > a,H: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ B2 )
= ( commut5242989786243415821_nat_a @ g @ addition @ zero @ H @ A2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_right
thf(fact_481_fincomp__mono__neutral__cong__right,axiom,
! [B2: set_a,A2: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ B2 )
= ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ H @ A2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_right
thf(fact_482_fincomp__mono__neutral__cong__left,axiom,
! [B2: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ A2 )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_left
thf(fact_483_fincomp__mono__neutral__cong__left,axiom,
! [B2: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ A2 )
= ( commut5242989786243415821_nat_a @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_left
thf(fact_484_fincomp__mono__neutral__cong__left,axiom,
! [B2: set_a,A2: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ A2 )
= ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong_left
thf(fact_485_card__le__sumset,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_486_card__sumset__0__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B2 )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_487_sum__list__Suc,axiom,
! [F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : ( suc @ ( F @ X3 ) )
@ Xs2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) ) @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% sum_list_Suc
thf(fact_488_sum__list__Suc,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X3: nat] : ( suc @ ( F @ X3 ) )
@ Xs2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ).
% sum_list_Suc
thf(fact_489_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_490_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_491_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_492_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_493_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_494_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_495_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_496_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_497_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_498_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_499_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_500_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_501_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_502_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_503_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_504_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_505_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_506_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_507_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_508_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_509_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_510_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_511_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_512_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_513_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_514_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_515_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_516_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_517_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_518_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_519_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_520_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_521_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_522_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_523_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_524_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_525_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_526_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_527_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_528_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_529_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_530_old_Onat_Osimps_I6_J,axiom,
! [F1: a,F22: nat > a > a] :
( ( rec_nat_a @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(6)
thf(fact_531_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_eq_nat @ N5 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_532_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_533_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_534_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_535_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_536_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_537_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_538_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_539_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_540_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_541_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_542_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_543_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_544_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_545_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_546_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_547_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_548_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_549_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_550_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_551_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_552_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_553_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_554_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_555_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_556_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_557_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_558_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_559_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_560_sum__list__0,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : zero_zero_nat
@ Xs2 ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_561_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_562_nth__minus__list,axiom,
! [I2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( minus_minus_list_nat @ Xs2 @ Ys2 ) @ I2 )
= ( minus_minus_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ).
% nth_minus_list
thf(fact_563_nth__minus__list,axiom,
! [I2: nat,Xs2: list_set_a,Ys2: list_set_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Ys2 ) )
=> ( ( nth_set_a @ ( minus_6559351004396594891_set_a @ Xs2 @ Ys2 ) @ I2 )
= ( minus_minus_set_a @ ( nth_set_a @ Xs2 @ I2 ) @ ( nth_set_a @ Ys2 @ I2 ) ) ) ) ) ).
% nth_minus_list
thf(fact_564_nth__minus__list,axiom,
! [I2: nat,Xs2: list_set_nat,Ys2: list_set_nat] :
( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Ys2 ) )
=> ( ( nth_set_nat @ ( minus_1998526526692677103et_nat @ Xs2 @ Ys2 ) @ I2 )
= ( minus_minus_set_nat @ ( nth_set_nat @ Xs2 @ I2 ) @ ( nth_set_nat @ Ys2 @ I2 ) ) ) ) ) ).
% nth_minus_list
thf(fact_565_Gmult__0,axiom,
! [A: a] :
( ( gmult_a @ addition @ zero @ A @ zero_zero_nat )
= zero ) ).
% Gmult_0
thf(fact_566_Gmult__1,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( gmult_a @ addition @ zero @ A @ ( suc @ zero_zero_nat ) )
= A ) ) ).
% Gmult_1
thf(fact_567_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_568_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_569_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_570_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_571_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_572_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_573_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_574_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_575_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_576_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_577_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_578_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_579_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_580_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_581_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_582_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_583_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_584_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_585_subgroup__transitive,axiom,
! [K4: set_a,H3: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_subgroup_a @ K4 @ H3 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H3 @ G2 @ Composition @ Unit )
=> ( group_subgroup_a @ K4 @ G2 @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_586_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_587_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_588_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_589_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_590_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_591_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_592_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_593_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_594_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_595_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_596_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_597_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_598_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_599_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_600_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_601_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_602_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_603_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_604_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M6: nat] :
( M7
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_605_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_606_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_607_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_608_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_609_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_610_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( R2 @ X2 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_611_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_612_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_613_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_614_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_615_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_616_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
& ( M4 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_617_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_618_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_619_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_620_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_621_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_622_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_623_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_624_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_625_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_626_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_627_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_628_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_629_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_630_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_631_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_632_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
? [K2: nat] :
( N5
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_633_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_634_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_635_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_636_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_637_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_638_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_639_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_640_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_641_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_642_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_643_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_644_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( P @ X2 @ Y4 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_645_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_646_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_647_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_648_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_649_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_650_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_651_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_652_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_653_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_654_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_655_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_656_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_657_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_658_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_659_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_660_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat_a,M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_subgroup_nat_a @ G2 @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ U @ G2 )
=> ( ( group_inverse_nat_a @ M2 @ Composition @ Unit @ U )
= ( group_inverse_nat_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_661_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat,M2: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_subgroup_nat @ G2 @ M2 @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( ( group_inverse_nat @ M2 @ Composition @ Unit @ U )
= ( group_inverse_nat @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_662_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_a,M2: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_subgroup_a @ G2 @ M2 @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( ( group_inverse_a @ M2 @ Composition @ Unit @ U )
= ( group_inverse_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_663_card__le__Suc0__iff__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_664_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_665_subgroup_Oaxioms_I2_J,axiom,
! [G2: set_a,M2: set_a,Composition: a > a > a,Unit: a] :
( ( group_subgroup_a @ G2 @ M2 @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ).
% subgroup.axioms(2)
thf(fact_666_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_667_minus__list__def,axiom,
( minus_6559351004396594891_set_a
= ( ^ [Xs: list_set_a,Ys: list_set_a] : ( map_Pr8866779738092938811_set_a @ ( produc4474087116711199794_set_a @ minus_minus_set_a ) @ ( zip_set_a_set_a @ Xs @ Ys ) ) ) ) ).
% minus_list_def
thf(fact_668_minus__list__def,axiom,
( minus_1998526526692677103et_nat
= ( ^ [Xs: list_set_nat,Ys: list_set_nat] : ( map_Pr6967709084026004015et_nat @ ( produc8983872132230954816et_nat @ minus_minus_set_nat ) @ ( zip_set_nat_set_nat @ Xs @ Ys ) ) ) ) ).
% minus_list_def
thf(fact_669_minus__list__def,axiom,
( minus_minus_list_nat
= ( ^ [Xs: list_nat,Ys: list_nat] : ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% minus_list_def
thf(fact_670_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_671_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_673_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_674_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_675_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_676_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_677_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_678_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_679_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_680_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_681_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_682_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_683_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_684_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_a,A2: set_nat_a,R: ( nat > a ) > ( nat > a ) > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
=> ? [B9: nat > a] :
( ( member_nat_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat > a,A22: nat > a,B3: nat > a] :
( ( member_nat_a @ A1 @ A2 )
=> ( ( member_nat_a @ A22 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_a @ A2 ) @ ( finite_card_nat_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_685_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_a,A2: set_a,R: a > ( nat > a ) > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: nat > a] :
( ( member_nat_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: nat > a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_686_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_a,A2: set_nat,R: nat > ( nat > a ) > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B9: nat > a] :
( ( member_nat_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat > a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_687_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat_a,R: ( nat > a ) > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat > a,A22: nat > a,B3: a] :
( ( member_nat_a @ A1 @ A2 )
=> ( ( member_nat_a @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_688_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_689_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_690_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_a,R: ( nat > a ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat > a,A22: nat > a,B3: nat] :
( ( member_nat_a @ A1 @ A2 )
=> ( ( member_nat_a @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_691_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_692_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_693_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_694_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_695_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_696_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_697_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_698_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_699_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_700_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_701_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_702_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_703_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_704_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_705_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_706_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_707_add__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 @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_708_add__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 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_709_add__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 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_710_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_711_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_712_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_713_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_714_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_715_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_716_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_717_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_718_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_719_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_720_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_721_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_722_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_723_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_724_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_725_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_726_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_727_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_728_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_729_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_730_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_731_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_732_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_733_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_734_card_Oeq__fold,axiom,
( finite_card_a
= ( finite_fold_a_nat
@ ^ [Uu: a] : suc
@ zero_zero_nat ) ) ).
% card.eq_fold
thf(fact_735_card_Oeq__fold,axiom,
( finite_card_nat
= ( finite_fold_nat_nat
@ ^ [Uu: nat] : suc
@ zero_zero_nat ) ) ).
% card.eq_fold
thf(fact_736_Khovanskii_OGmult__0,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( gmult_a @ Addition @ Zero @ A @ zero_zero_nat )
= Zero ) ) ).
% Khovanskii.Gmult_0
thf(fact_737_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat_a,M2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a] :
( ( group_subgroup_nat_a @ G2 @ M2 @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M2 @ Composition @ Unit @ X )
=> ( ( member_nat_a @ X @ M2 )
=> ( ( member_nat_a @ ( group_inverse_nat_a @ M2 @ Composition @ Unit @ X ) @ G2 )
= ( member_nat_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_738_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat,M2: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G2 @ M2 @ Composition @ Unit )
=> ( ( group_invertible_nat @ M2 @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M2 )
=> ( ( member_nat @ ( group_inverse_nat @ M2 @ Composition @ Unit @ X ) @ G2 )
= ( member_nat @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_739_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_a,M2: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G2 @ M2 @ Composition @ Unit )
=> ( ( group_invertible_a @ M2 @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M2 )
=> ( ( member_a @ ( group_inverse_a @ M2 @ Composition @ Unit @ X ) @ G2 )
= ( member_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_740_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_741_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_742_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_743_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_744_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_745_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_746_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B8: set_a] :
( ( ord_less_eq_set_a @ A2 @ B8 )
& ( ord_less_eq_set_a @ B8 @ C2 )
& ( ( finite_card_a @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_747_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_748_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S2 )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_749_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_750_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G5: set_a] :
( ( ord_less_eq_set_a @ G5 @ F2 )
=> ( ( finite_finite_a @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G5 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_751_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G5: set_nat] :
( ( ord_less_eq_set_nat @ G5 @ F2 )
=> ( ( finite_finite_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G5 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_752_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_753_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_754_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_755_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_756_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_757_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_758_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_759_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_760_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_761_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_762_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_763_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_764_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_765_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_766_Khovanskii_OGmult__1,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( ( gmult_nat_a @ Addition @ Zero @ A @ ( suc @ zero_zero_nat ) )
= A ) ) ) ).
% Khovanskii.Gmult_1
thf(fact_767_Khovanskii_OGmult__1,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ G2 )
=> ( ( gmult_nat @ Addition @ Zero @ A @ ( suc @ zero_zero_nat ) )
= A ) ) ) ).
% Khovanskii.Gmult_1
thf(fact_768_Khovanskii_OGmult__1,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ G2 )
=> ( ( gmult_a @ Addition @ Zero @ A @ ( suc @ zero_zero_nat ) )
= A ) ) ) ).
% Khovanskii.Gmult_1
thf(fact_769_card__less,axiom,
! [M2: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_770_card__less__Suc,axiom,
! [M2: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_771_card__less__Suc2,axiom,
! [M2: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M2 )
& ( ord_less_nat @ K2 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M2 )
& ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_772_commutative__monoid_Ofincomp__mono__neutral__cong__right,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat,A2: set_nat,G: nat > a,H: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ B2 )
= ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ H @ A2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_right
thf(fact_773_commutative__monoid_Ofincomp__mono__neutral__cong__right,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat_a,A2: set_nat_a,G: ( nat > a ) > a,H: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ B2 )
= ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ H @ A2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_right
thf(fact_774_commutative__monoid_Ofincomp__mono__neutral__cong__right,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_a,A2: set_a,G: a > a,H: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ B2 )
= ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ H @ A2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_right
thf(fact_775_commutative__monoid_Ofincomp__mono__neutral__cong__left,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_left
thf(fact_776_commutative__monoid_Ofincomp__mono__neutral__cong__left,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_left
thf(fact_777_commutative__monoid_Ofincomp__mono__neutral__cong__left,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_a,A2: set_a,H: a > a,G: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong_left
thf(fact_778_commutative__monoid_Ofincomp__Suc2,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( Composition
@ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ) ).
% commutative_monoid.fincomp_Suc2
thf(fact_779_sum__list__addf,axiom,
! [F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : ( plus_plus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_Pr3938374229010428429at_nat @ G @ Xs2 ) ) ) ) ).
% sum_list_addf
thf(fact_780_sum__list__mono2,axiom,
! [Xs2: list_real,Ys2: list_real] :
( ( ( size_size_list_real @ Xs2 )
= ( size_size_list_real @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
=> ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Ys2 @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6723090944982001619t_real @ Xs2 ) @ ( groups6723090944982001619t_real @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_781_sum__list__mono2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_782_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_783_Ruzsa__triangle__ineq1,axiom,
! [U3: set_a,V4: set_a,W: set_a] :
( ( finite_finite_a @ U3 )
=> ( ( ord_less_eq_set_a @ U3 @ g )
=> ( ( finite_finite_a @ V4 )
=> ( ( ord_less_eq_set_a @ V4 @ g )
=> ( ( finite_finite_a @ W )
=> ( ( ord_less_eq_set_a @ W @ g )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ U3 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ V4 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ V4 ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W ) ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq1
thf(fact_784_fincomp__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : g ) )
=> ( ( addition @ ( F @ zero_zero_nat ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% fincomp_0'
thf(fact_785_sumset__iterated__empty,axiom,
! [R: nat] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ).
% sumset_iterated_empty
thf(fact_786_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_787_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_788_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_789_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_790_fincomp__mono__neutral__cong,axiom,
! [B2: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ A2 )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong
thf(fact_791_fincomp__mono__neutral__cong,axiom,
! [B2: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ A2 )
= ( commut5242989786243415821_nat_a @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong
thf(fact_792_fincomp__mono__neutral__cong,axiom,
! [B2: set_a,A2: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= zero ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( ( G @ I3 )
= zero ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ A2 )
= ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% fincomp_mono_neutral_cong
thf(fact_793_nonempty,axiom,
a2 != bot_bot_set_a ).
% nonempty
thf(fact_794_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_795_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_796_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_797_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_798_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_799_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_800_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_801_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_802_finite__sumset_H,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset'
thf(fact_803_card__sumset__0__iff_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B2 @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_804_infinite__sumset__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_805_infinite__sumset__aux,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_806_Gmult__diff,axiom,
! [A: a,N: nat,M: nat] :
( ( member_a @ A @ g )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( addition @ ( gmult_a @ addition @ zero @ A @ M ) @ ( group_inverse_a @ g @ addition @ zero @ ( gmult_a @ addition @ zero @ A @ N ) ) )
= ( gmult_a @ addition @ zero @ A @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% Gmult_diff
thf(fact_807_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_808_atLeastAtMost__iff,axiom,
! [I2: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_809_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_810_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_811_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H4: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H4 ) )
= ( ( ( L = L2 )
& ( H = H4 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H4 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_812_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H4: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H4 ) )
= ( ( ( L = L2 )
& ( H = H4 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H4 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_813_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H4: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H4 ) )
= ( ( ( L = L2 )
& ( H = H4 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H4 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_814_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H4: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H4 ) )
= ( ( ( L = L2 )
& ( H = H4 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H4 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_815_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_816_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_817_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_818_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_819_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_820_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_821_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_822_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_823_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_824_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_825_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_826_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_827_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_828_funcset__Int__left,axiom,
! [F: nat > a,A2: set_nat,C2: set_a,B2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ ( inf_inf_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Int_left
thf(fact_829_atLeastatMost__empty__iff2,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_830_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_831_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_832_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_833_atLeastatMost__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( set_or6288561110385358355_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_834_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_835_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_836_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_837_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_838_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_839_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_840_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_841_atLeastatMost__empty,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( ( set_or6836045993805503595st_nat @ A @ B )
= bot_bot_set_list_nat ) ) ).
% atLeastatMost_empty
thf(fact_842_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_843_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_844_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_845_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_846_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_847_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_848_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_849_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_850_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_851_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_852_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_853_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_854_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_855_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_856_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_857_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_858_Icc__subset__Iic__iff,axiom,
! [L: set_a,H: set_a,H4: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_ord_atMost_set_a @ H4 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ H @ H4 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_859_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H: set_nat,H4: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or4236626031148496127et_nat @ H4 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H )
| ( ord_less_eq_set_nat @ H @ H4 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_860_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H4: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H4 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H4 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_861_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H4: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H4 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H4 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_862_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_863_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_864_sumset__Int__carrier,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier
thf(fact_865_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B2 @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_866_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_867_sumset__is__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B2 @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_868_local_Osumset__empty_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% local.sumset_empty(1)
thf(fact_869_local_Osumset__empty_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% local.sumset_empty(2)
thf(fact_870_minus__minusset,axiom,
! [A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% minus_minusset
thf(fact_871_fincomp__empty,axiom,
! [F: nat > a] :
( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ bot_bot_set_nat )
= zero ) ).
% fincomp_empty
thf(fact_872_fincomp__empty,axiom,
! [F: ( nat > a ) > a] :
( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ F @ bot_bot_set_nat_a )
= zero ) ).
% fincomp_empty
thf(fact_873_fincomp__empty,axiom,
! [F: a > a] :
( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ F @ bot_bot_set_a )
= zero ) ).
% fincomp_empty
thf(fact_874_card__minusset,axiom,
! [A2: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) ).
% card_minusset
thf(fact_875_minusset__is__empty__iff,axiom,
! [A2: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_876_lambda__zero,axiom,
( ( ^ [H5: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_877_lambda__zero,axiom,
( ( ^ [H5: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_889_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_890_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_891_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_892_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_893_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_894_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_895_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_896_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_897_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_898_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_899_ivl__disj__int__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(4)
thf(fact_900_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_901_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_902_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_903_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_904_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_905_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_906_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_907_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_908_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_909_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_910_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_911_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_912_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_913_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_914_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_915_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_916_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_917_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_918_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_919_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_920_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_921_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_922_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_923_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_924_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_925_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_926_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_927_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_928_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_952_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_953_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_954_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_955_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_956_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_957_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_958_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_959_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_960_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_961_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_962_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_963_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_964_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_965_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_966_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_967_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_968_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_969_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_970_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_971_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_972_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_973_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_974_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_975_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_976_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_977_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_978_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_979_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_980_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_981_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_982_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_983_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_984_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_985_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_986_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_987_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_988_card__Diff__subset__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_989_card__Diff__subset__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_990_Khovanskii_Ononempty,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( A2 != bot_bot_set_a ) ) ).
% Khovanskii.nonempty
thf(fact_991_Khovanskii_Ononempty,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( A2 != bot_bot_set_nat ) ) ).
% Khovanskii.nonempty
thf(fact_992_infinite__Icc,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_993_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_994_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_995_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_996_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_997_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_998_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_999_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1000_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1001_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1002_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1003_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1004_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1005_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1006_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1007_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1008_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1009_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1010_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1011_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1012_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1013_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1014_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1015_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1016_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1017_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1018_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_1019_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1020_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1021_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1022_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1023_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1024_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_1025_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1026_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1027_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1028_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_1029_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_1030_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1031_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_1032_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1033_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1034_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1035_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1036_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1037_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1038_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_1039_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1040_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1041_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1042_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1043_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1044_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1045_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1046_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1047_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1048_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1049_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1050_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1051_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1052_commutative__monoid_Ofincomp__empty,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ bot_bot_set_nat )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_1053_commutative__monoid_Ofincomp__empty,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ F @ bot_bot_set_nat_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_1054_commutative__monoid_Ofincomp__empty,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ F @ bot_bot_set_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_1055_sum__list__const__mult,axiom,
! [C: nat,F: product_prod_nat_nat > nat,Xs2: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : ( times_times_nat @ C @ ( F @ X3 ) )
@ Xs2 ) )
= ( times_times_nat @ C @ ( groups4561878855575611511st_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) ) ) ) ).
% sum_list_const_mult
thf(fact_1056_sum__list__mult__const,axiom,
! [F: product_prod_nat_nat > nat,C: nat,Xs2: list_P6011104703257516679at_nat] :
( ( groups4561878855575611511st_nat
@ ( map_Pr3938374229010428429at_nat
@ ^ [X3: product_prod_nat_nat] : ( times_times_nat @ ( F @ X3 ) @ C )
@ Xs2 ) )
= ( times_times_nat @ ( groups4561878855575611511st_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs2 ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_1057_atLeastatMost__psubset__iff,axiom,
! [A: list_nat,B: list_nat,C: list_nat,D: list_nat] :
( ( ord_le1190675801316882794st_nat @ ( set_or6836045993805503595st_nat @ A @ B ) @ ( set_or6836045993805503595st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_list_nat @ A @ B )
| ( ( ord_less_eq_list_nat @ C @ A )
& ( ord_less_eq_list_nat @ B @ D )
& ( ( ord_less_list_nat @ C @ A )
| ( ord_less_list_nat @ B @ D ) ) ) )
& ( ord_less_eq_list_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1058_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1059_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1060_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1061_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1062_commutative__monoid_Ofuncset__Int__left,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat,C2: set_a,B2: set_nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ ( inf_inf_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C2 ) ) ) ) ) ).
% commutative_monoid.funcset_Int_left
thf(fact_1063_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1064_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1065_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1066_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_1067_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1068_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1069_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1070_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1071_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1072_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1073_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1074_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1075_diff__card__le__card__Diff,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1076_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1077_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1078_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1079_Khovanskii_OGmult__diff,axiom,
! [G2: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: set_nat_a,A: nat > a,N: nat,M: nat] :
( ( khovanskii_nat_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat_a @ A @ G2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( Addition @ ( gmult_nat_a @ Addition @ Zero @ A @ M ) @ ( group_inverse_nat_a @ G2 @ Addition @ Zero @ ( gmult_nat_a @ Addition @ Zero @ A @ N ) ) )
= ( gmult_nat_a @ Addition @ Zero @ A @ ( minus_minus_nat @ M @ N ) ) ) ) ) ) ).
% Khovanskii.Gmult_diff
thf(fact_1080_Khovanskii_OGmult__diff,axiom,
! [G2: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,N: nat,M: nat] :
( ( khovanskii_nat @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_nat @ A @ G2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( Addition @ ( gmult_nat @ Addition @ Zero @ A @ M ) @ ( group_inverse_nat @ G2 @ Addition @ Zero @ ( gmult_nat @ Addition @ Zero @ A @ N ) ) )
= ( gmult_nat @ Addition @ Zero @ A @ ( minus_minus_nat @ M @ N ) ) ) ) ) ) ).
% Khovanskii.Gmult_diff
thf(fact_1081_Khovanskii_OGmult__diff,axiom,
! [G2: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,N: nat,M: nat] :
( ( khovanskii_a @ G2 @ Addition @ Zero @ A2 )
=> ( ( member_a @ A @ G2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( Addition @ ( gmult_a @ Addition @ Zero @ A @ M ) @ ( group_inverse_a @ G2 @ Addition @ Zero @ ( gmult_a @ Addition @ Zero @ A @ N ) ) )
= ( gmult_a @ Addition @ Zero @ A @ ( minus_minus_nat @ M @ N ) ) ) ) ) ) ).
% Khovanskii.Gmult_diff
thf(fact_1082_commutative__monoid_Ofincomp__mono__neutral__cong,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong
thf(fact_1083_commutative__monoid_Ofincomp__mono__neutral__cong,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong
thf(fact_1084_commutative__monoid_Ofincomp__mono__neutral__cong,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,B2: set_a,A2: set_a,H: a > a,G: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B2 @ A2 ) )
=> ( ( H @ I3 )
= Unit ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( ( G @ I3 )
= Unit ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ A2 )
= ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ H @ B2 ) ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_mono_neutral_cong
thf(fact_1085_commutative__monoid_Ofincomp__0_H,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : M2 ) )
=> ( ( Composition @ ( F @ zero_zero_nat ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% commutative_monoid.fincomp_0'
thf(fact_1086_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1087_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1088_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1089_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1090_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1091_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1092_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1093_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_1094_sumset__iterated__r,axiom,
! [R: nat,A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_1095_fincomp__Un__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( addition @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_1096_fincomp__Un__disjoint,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) )
= ( addition @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_1097_fincomp__Un__disjoint,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( addition @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_1098_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_1099_sumset__subset__Un1,axiom,
! [A2: set_a,A7: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A7 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ B2 ) ) ) ).
% sumset_subset_Un1
thf(fact_1100_sumset__subset__Un2,axiom,
! [A2: set_a,B2: set_a,B6: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ B6 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B6 ) ) ) ).
% sumset_subset_Un2
thf(fact_1101_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B2: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C2 ) @ B2 ) ) ).
% sumset_subset_Un(2)
thf(fact_1102_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B2: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sumset_subset_Un(1)
thf(fact_1103_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1104_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1105_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1106_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1107_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_1108_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_1109_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1110_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_1111_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_1112_funcset__Un__left,axiom,
! [F: nat > a,A2: set_nat,B2: set_nat,C2: set_a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( sup_sup_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C2 ) )
= ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C2 ) )
& ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Un_left
thf(fact_1113_fincomp__Un__Int,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : g ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : g ) )
=> ( ( addition @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( addition @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1114_fincomp__Un__Int,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : g ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : g ) )
=> ( ( addition @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) )
= ( addition @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1115_fincomp__Un__Int,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : g ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : g ) )
=> ( ( addition @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ ( inf_inf_set_a @ A2 @ B2 ) ) )
= ( addition @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ g @ addition @ zero @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1116_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1117_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1118_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1119_Ruzsa__triangle__ineq2,axiom,
! [U3: set_a,V4: set_a,W: set_a] :
( ( finite_finite_a @ U3 )
=> ( ( ord_less_eq_set_a @ U3 @ g )
=> ( ( U3 != bot_bot_set_a )
=> ( ( finite_finite_a @ V4 )
=> ( ( ord_less_eq_set_a @ V4 @ g )
=> ( ( finite_finite_a @ W )
=> ( ( ord_less_eq_set_a @ W @ g )
=> ( ord_less_eq_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V4 @ W ) @ ( times_times_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V4 @ U3 ) @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ U3 @ W ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq2
thf(fact_1120_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1121_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1122_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1123_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1124_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1125_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_1126_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1127_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1128_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1129_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1130_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1131_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1132_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1133_Un__infinite,axiom,
! [S2: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1134_Un__infinite,axiom,
! [S2: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1135_infinite__Un,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S2 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1136_infinite__Un,axiom,
! [S2: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_a @ S2 )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_1137_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1138_ivl__disj__un__two__touch_I4_J,axiom,
! [L: real,M: real,U: real] :
( ( ord_less_eq_real @ L @ M )
=> ( ( ord_less_eq_real @ M @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M ) @ ( set_or1222579329274155063t_real @ M @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1139_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1140_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1141_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1142_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1143_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1144_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1145_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1146_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1147_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1148_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1149_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1150_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1151_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1152_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1153_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1154_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1155_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1156_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1157_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1158_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1159_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1160_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_1161_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1162_commutative__monoid_Ofuncset__Un__left,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat,B2: set_nat,C2: set_a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( sup_sup_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C2 ) )
= ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C2 ) )
& ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) ) ) ) ) ).
% commutative_monoid.funcset_Un_left
thf(fact_1163_card__Un__le,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ).
% card_Un_le
thf(fact_1164_card__Un__le,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ).
% card_Un_le
thf(fact_1165_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1166_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1167_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1168_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1169_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1170_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1171_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1172_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1173_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1174_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1175_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1176_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1177_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N5: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1178_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N5: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1179_card__Un__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1180_card__Un__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
= ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1181_ivl__disj__un__one_I4_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or5984915006950818249n_real @ L ) @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( set_ord_atMost_real @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_1182_ivl__disj__un__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_1183_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1184_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1185_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1186_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1187_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1188_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1189_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1190_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1191_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V2: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1192_card__Un__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( finite_card_a @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1193_card__Un__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1194_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V2: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1195_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ( Composition @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( Composition @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1196_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( Composition @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) )
= ( Composition @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1197_commutative__monoid_Ofincomp__Un__Int,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ( Composition @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ ( inf_inf_set_a @ A2 @ B2 ) ) )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_Int
thf(fact_1198_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M2 ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M2 ) )
=> ( ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( Composition @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1199_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M2 ) )
=> ( ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) )
= ( Composition @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1200_commutative__monoid_Ofincomp__Un__disjoint,axiom,
! [M2: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M2 @ Composition @ Unit )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M2 ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M2 ) )
=> ( ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ M2 @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Un_disjoint
thf(fact_1201_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1202_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_1203_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_1204_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1205_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1206_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1207_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1208_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1209_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1210_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_1211_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1212_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1213_card__sumset__singleton__eq,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_1214_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1215_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1216_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1217_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1218_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1219_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1220_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1221_sumsetdiff__sing,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_1222_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% sumset_subset_insert(1)
thf(fact_1223_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B2 ) ) ).
% sumset_subset_insert(2)
thf(fact_1224_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_1225_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1226_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_1227_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_1228_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a @ X @ B2 )
& ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1229_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1230_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1231_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_1232_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_1233_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1234_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1235_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1236_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1237_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1238_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1239_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1240_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1241_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y2: a,Z: a] : ( Y2 = Z )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_1242_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1243_card__sumset__le,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_1244_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1245_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1246_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1247_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1248_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1249_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_1250_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1251_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1252_card__insert__disjoint,axiom,
! [A2: set_nat_a,X: nat > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ~ ( member_nat_a @ X @ A2 )
=> ( ( finite_card_nat_a @ ( insert_nat_a @ X @ A2 ) )
= ( suc @ ( finite_card_nat_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1253_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1254_card__insert__disjoint,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1255_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1256_minusset__triv,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% minusset_triv
thf(fact_1257_sumset__iterated__0,axiom,
! [A2: set_a] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ zero_zero_nat )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% sumset_iterated_0
thf(fact_1258_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_1259_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_1260_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1261_fincomp__0,axiom,
! [F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat )
@ ^ [Uu: nat] : g ) )
=> ( ( commut6741328216151336360_a_nat @ g @ addition @ zero @ F @ ( set_ord_atMost_nat @ zero_zero_nat ) )
= ( F @ zero_zero_nat ) ) ) ).
% fincomp_0
thf(fact_1262_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1263_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1264_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1265_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1266_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1267_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1268_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_1269_Plu__2__2,axiom,
! [A0: set_a,B2: set_a,K0: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A0 @ B2 ) ) ) @ ( times_times_real @ K0 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A0 ) ) ) )
=> ( ( finite_finite_a @ A0 )
=> ( ( ord_less_eq_set_a @ A0 @ g )
=> ( ( A0 != bot_bot_set_a )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ( B2 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A0 )
=> ( ( A6 != bot_bot_set_a )
=> ! [K5: real] :
( ( ord_less_real @ zero_zero_real @ K5 )
=> ( ( ord_less_eq_real @ K5 @ K0 )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ g )
=> ( ( finite_finite_a @ C5 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C5 ) ) ) ) @ ( times_times_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ C5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Plu_2_2
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $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,
( ( alpha_a @ g @ addition @ zero @ a2 @ ( plus_plus_list_nat @ x @ y ) )
= ( commut6741328216151336360_a_nat @ g @ addition @ zero
@ ^ [I: nat] : ( gmult_a @ addition @ zero @ ( nth_a @ ( aA_a @ a2 ) @ I ) @ ( nth_nat @ ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ plus_plus_nat ) @ ( zip_nat_nat @ x @ y ) ) @ I ) )
@ ( set_ord_lessThan_nat @ ( finite_card_a @ a2 ) ) ) ) ).
%------------------------------------------------------------------------------