TPTP Problem File: SLH0917^1.p
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- 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_01211_047357__13734370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1505 ( 458 unt; 218 typ; 0 def)
% Number of atoms : 3900 (1173 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10494 ( 353 ~; 41 |; 144 &;7959 @)
% ( 0 <=>;1997 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 857 ( 857 >; 0 *; 0 +; 0 <<)
% Number of symbols : 201 ( 198 usr; 27 con; 0-3 aty)
% Number of variables : 3767 ( 228 ^;3424 !; 115 ?;3767 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:16:05.066
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
set_se7082751667426497518st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
set_se5258582372428582328st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $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_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__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (198)
thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: real > nat > real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Int__Oint,type,
complete_Inf_Inf_int: set_int > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comple184543376406953807st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
comple8462666950445340293st_nat: set_set_set_list_nat > set_set_list_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
comple5189992959352112827st_nat: set_se5258582372428582328st_nat > set_set_set_list_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
comple8180622878413557745st_nat: set_se7082751667426497518st_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple9105089376463352645_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Countable__Set_Oto__nat__on_001tf__a,type,
counta3566351752493190365t_on_a: set_a > a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
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_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on8624761805129053417st_nat: ( list_nat > set_list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
inj_on_list_nat_a: ( list_nat > a ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on1516391927397616473at_nat: ( set_list_nat > nat ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
inj_on211238342484403029st_nat: ( set_list_nat > set_set_list_nat ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001tf__a,type,
inj_on7310907584571128757_nat_a: ( set_list_nat > a ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_a_list_nat: ( a > list_nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on6747084356327149539st_nat: ( a > set_list_nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__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__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
minus_1490147023502258853st_nat: set_set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
minus_7520675481749758683st_nat: set_set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__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_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
if_set_list_nat: $o > set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
if_set_set_list_nat: $o > set_set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
if_set7539935375068789052st_nat: $o > set_set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
if_set4742882518001122930st_nat: $o > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001tf__a,type,
aA_a: set_a > list_a ).
thf(sy_c_Khovanskii_OKhovanskii_Oaugmentum,type,
augmentum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Odementum,type,
dementum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olength__sum__set,type,
length_sum_set: nat > nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olist__incr,type,
list_incr: nat > list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elements,type,
minimal_elements: set_list_nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elementsp,type,
minimal_elementsp: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Khovanskii_Omax__pointwise,type,
max_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Omin__pointwise,type,
min_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_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__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo7146503454104809893_nat_o: set_list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_M_Eo_J,type,
bot_bo6889798525461962991_nat_o: set_set_list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
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__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo3886227569956363488st_nat: set_set_list_nat ).
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_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
bot_bo3499706412017099030st_nat: set_set_set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
bot_bo1158166727579713100st_nat: set_se5258582372428582328st_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
bot_bo3510429811944912002st_nat: set_se7082751667426497518st_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__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le6760893623726761120st_nat: set_set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
ord_le1759137698939693270st_nat: set_set_set_list_nat > set_set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
ord_le3416919459018737164st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le1068707526560357548st_nat: set_set_list_nat > set_set_list_nat > $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_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
ord_le7100322305783427298st_nat: set_set_set_list_nat > set_set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
ord_le2499698639687704088st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
ord_le647588107330380878st_nat: set_se7082751667426497518st_nat > set_se7082751667426497518st_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_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_max_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_ma6513994572189496503st_nat: set_set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
ord_ma3801742943350824173st_nat: set_set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
ord_ma1726305516211428899st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_Itf__a_J,type,
ord_max_set_a: set_a > set_a > set_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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
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_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
collec4691811733418234273st_nat: ( set_set_list_nat > $o ) > set_set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
collec696328324557263319st_nat: ( set_set_set_list_nat > $o ) > set_se5258582372428582328st_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_OPow_001t__List__Olist_It__Nat__Onat_J,type,
pow_list_nat: set_list_nat > set_set_list_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
pow_set_list_nat: set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_8532145185254316925st_nat: ( list_nat > set_list_nat ) > set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
image_list_nat_a: ( list_nat > a ) > set_list_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_2883343038133793645st_nat: ( nat > set_list_nat ) > set_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_3522889415257180889st_nat: ( nat > set_set_set_list_nat ) > set_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
image_6410155499156649085st_nat: ( set_list_nat > list_nat ) > set_set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1475345664228658157at_nat: ( set_list_nat > nat ) > set_set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_920372530894776809st_nat: ( set_list_nat > set_set_list_nat ) > set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001tf__a,type,
image_set_list_nat_a: ( set_list_nat > a ) > set_set_list_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
image_set_nat_int: ( set_nat > int ) > set_set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
image_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001tf__a,type,
image_8692725287357501035_nat_a: ( set_set_list_nat > a ) > set_set_set_list_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
image_a_list_nat: ( a > list_nat ) > set_a > set_list_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Real__Oreal,type,
image_a_real: ( a > real ) > set_a > set_real ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_a_set_list_nat: ( a > set_list_nat ) > set_a > set_set_list_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_6865978040264483717st_nat: ( a > set_set_list_nat ) > set_a > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_5164723790120178619st_nat: ( a > set_set_set_list_nat ) > set_a > set_se5258582372428582328st_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
insert_set_list_nat: set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
insert721039030076969596st_nat: set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Weierstrass__Theorems_Oreal__polynomial__function_001t__Real__Oreal,type,
weiers3457258110322917882n_real: ( real > real ) > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $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_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
member1029098694177496419st_nat: set_set_list_nat > set_set_set_list_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
member7304678173793621401st_nat: set_set_set_list_nat > set_se5258582372428582328st_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
member4920245652161177295st_nat: set_se5258582372428582328st_nat > set_se7082751667426497518st_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $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_C____,type,
c: nat > list_nat > set_list_nat ).
thf(sy_v_SUM1____,type,
sUM1: int ).
thf(sy_v_SUM2____,type,
sUM2: int ).
thf(sy_v_U____,type,
u: nat > set_list_nat ).
thf(sy_v_X____,type,
x: set_list_nat ).
thf(sy_v_Y____,type,
y: set_list_nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p0____,type,
p0: real > real ).
thf(sy_v_q____,type,
q: real > real ).
thf(sy_v_r____,type,
r: nat ).
% Relevant facts (1267)
thf(fact_0_that_I2_J,axiom,
y != bot_bot_set_list_nat ).
% that(2)
thf(fact_1_that_I1_J,axiom,
ord_le6045566169113846134st_nat @ y @ x ).
% that(1)
thf(fact_2_Inf__superset__mono,axiom,
! [B: set_se7082751667426497518st_nat,A: set_se7082751667426497518st_nat] :
( ( ord_le647588107330380878st_nat @ B @ A )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ ( comple8180622878413557745st_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_3_Inf__superset__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_4_Inf__superset__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_5_Inf__superset__mono,axiom,
! [B: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_6_Inf__superset__mono,axiom,
! [B: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_7_Inf__superset__mono,axiom,
! [B: set_set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_8_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_list_nat,F: a > set_list_nat] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_9_INF__eq,axiom,
! [A: set_a,B: set_nat,G: nat > set_list_nat,F: a > set_list_nat] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_10_INF__eq,axiom,
! [A: set_nat,B: set_a,G: a > set_list_nat,F: nat > set_list_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_11_INF__eq,axiom,
! [A: set_nat,B: set_nat,G: nat > set_list_nat,F: nat > set_list_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_12_INF__eq,axiom,
! [A: set_list_nat,B: set_a,G: a > set_list_nat,F: list_nat > set_list_nat] :
( ! [I: list_nat] :
( ( member_list_nat @ I @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_13_INF__eq,axiom,
! [A: set_list_nat,B: set_nat,G: nat > set_list_nat,F: list_nat > set_list_nat] :
( ! [I: list_nat] :
( ( member_list_nat @ I @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_14_INF__eq,axiom,
! [A: set_a,B: set_list_nat,G: list_nat > set_list_nat,F: a > set_list_nat] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X: list_nat] :
( ( member_list_nat @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: list_nat] :
( ( member_list_nat @ J @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_15_INF__eq,axiom,
! [A: set_nat,B: set_list_nat,G: list_nat > set_list_nat,F: nat > set_list_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X: list_nat] :
( ( member_list_nat @ X @ B )
& ( ord_le6045566169113846134st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: list_nat] :
( ( member_list_nat @ J @ B )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_16_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_set_set_list_nat,F: a > set_set_set_list_nat] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_le7100322305783427298st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_le7100322305783427298st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple5189992959352112827st_nat @ ( image_5164723790120178619st_nat @ F @ A ) )
= ( comple5189992959352112827st_nat @ ( image_5164723790120178619st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_17_INF__eq,axiom,
! [A: set_a,B: set_nat,G: nat > set_set_set_list_nat,F: a > set_set_set_list_nat] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_le7100322305783427298st_nat @ ( G @ X ) @ ( F @ I ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_le7100322305783427298st_nat @ ( F @ X ) @ ( G @ J ) ) ) )
=> ( ( comple5189992959352112827st_nat @ ( image_5164723790120178619st_nat @ F @ A ) )
= ( comple5189992959352112827st_nat @ ( image_3522889415257180889st_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_18_InterI,axiom,
! [C: set_set_set_nat,A: set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ C )
=> ( member_set_nat @ A @ X2 ) )
=> ( member_set_nat @ A @ ( comple1065008630642458357et_nat @ C ) ) ) ).
% InterI
thf(fact_19_InterI,axiom,
! [C: set_set_set_a,A: set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C )
=> ( member_set_a @ A @ X2 ) )
=> ( member_set_a @ A @ ( comple9105089376463352645_set_a @ C ) ) ) ).
% InterI
thf(fact_20_InterI,axiom,
! [C: set_set_set_list_nat,A: set_list_nat] :
( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ C )
=> ( member_set_list_nat @ A @ X2 ) )
=> ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C ) ) ) ).
% InterI
thf(fact_21_InterI,axiom,
! [C: set_se5258582372428582328st_nat,A: set_set_list_nat] :
( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ C )
=> ( member1029098694177496419st_nat @ A @ X2 ) )
=> ( member1029098694177496419st_nat @ A @ ( comple5189992959352112827st_nat @ C ) ) ) ).
% InterI
thf(fact_22_InterI,axiom,
! [C: set_set_a,A: a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( member_a @ A @ X2 ) )
=> ( member_a @ A @ ( comple6135023378680113637_set_a @ C ) ) ) ).
% InterI
thf(fact_23_InterI,axiom,
! [C: set_set_nat,A: nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
=> ( member_nat @ A @ X2 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) ) ) ).
% InterI
thf(fact_24_InterI,axiom,
! [C: set_set_list_nat,A: list_nat] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ C )
=> ( member_list_nat @ A @ X2 ) )
=> ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C ) ) ) ).
% InterI
thf(fact_25_Inter__iff,axiom,
! [A: set_nat,C: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple1065008630642458357et_nat @ C ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ C )
=> ( member_set_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_26_Inter__iff,axiom,
! [A: set_a,C: set_set_set_a] :
( ( member_set_a @ A @ ( comple9105089376463352645_set_a @ C ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ C )
=> ( member_set_a @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_27_Inter__iff,axiom,
! [A: set_list_nat,C: set_set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C ) )
= ( ! [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ C )
=> ( member_set_list_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_28_Inter__iff,axiom,
! [A: set_set_list_nat,C: set_se5258582372428582328st_nat] :
( ( member1029098694177496419st_nat @ A @ ( comple5189992959352112827st_nat @ C ) )
= ( ! [X3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X3 @ C )
=> ( member1029098694177496419st_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_29_Inter__iff,axiom,
! [A: a,C: set_set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C )
=> ( member_a @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_30_Inter__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( member_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_31_Inter__iff,axiom,
! [A: list_nat,C: set_set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ C )
=> ( member_list_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_32_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_34_subsetI,axiom,
! [A: set_set_list_nat,B: set_set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( member_set_list_nat @ X4 @ B ) )
=> ( ord_le1068707526560357548st_nat @ A @ B ) ) ).
% subsetI
thf(fact_35_subsetI,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ A )
=> ( member7304678173793621401st_nat @ X4 @ B ) )
=> ( ord_le2499698639687704088st_nat @ A @ B ) ) ).
% subsetI
thf(fact_36_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_37_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_38_subsetI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_list_nat @ X4 @ B ) )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( member1029098694177496419st_nat @ X4 @ B ) )
=> ( ord_le7100322305783427298st_nat @ A @ B ) ) ).
% subsetI
thf(fact_40_subset__antisym,axiom,
! [A: set_set_list_nat,B: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B )
=> ( ( ord_le1068707526560357548st_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_41_subset__antisym,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ B )
=> ( ( ord_le2499698639687704088st_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_42_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_43_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_44_subset__antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_45_subset__antisym,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B )
=> ( ( ord_le7100322305783427298st_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_46_image__eqI,axiom,
! [B2: a,F: a > a,X5: a,A: set_a] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_47_image__eqI,axiom,
! [B2: nat,F: a > nat,X5: a,A: set_a] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_48_image__eqI,axiom,
! [B2: a,F: nat > a,X5: nat,A: set_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_49_image__eqI,axiom,
! [B2: nat,F: nat > nat,X5: nat,A: set_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_50_image__eqI,axiom,
! [B2: a,F: list_nat > a,X5: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_list_nat @ X5 @ A )
=> ( member_a @ B2 @ ( image_list_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_51_image__eqI,axiom,
! [B2: nat,F: list_nat > nat,X5: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_list_nat @ X5 @ A )
=> ( member_nat @ B2 @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_52_image__eqI,axiom,
! [B2: list_nat,F: a > list_nat,X5: a,A: set_a] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A )
=> ( member_list_nat @ B2 @ ( image_a_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_53_image__eqI,axiom,
! [B2: list_nat,F: nat > list_nat,X5: nat,A: set_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A )
=> ( member_list_nat @ B2 @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_54_image__eqI,axiom,
! [B2: list_nat,F: list_nat > list_nat,X5: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_list_nat @ X5 @ A )
=> ( member_list_nat @ B2 @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_55_image__eqI,axiom,
! [B2: set_list_nat,F: list_nat > set_list_nat,X5: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X5 ) )
=> ( ( member_list_nat @ X5 @ A )
=> ( member_set_list_nat @ B2 @ ( image_8532145185254316925st_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_56_order__refl,axiom,
! [X5: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_57_order__refl,axiom,
! [X5: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_58_order__refl,axiom,
! [X5: set_nat] : ( ord_less_eq_set_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_59_order__refl,axiom,
! [X5: set_a] : ( ord_less_eq_set_a @ X5 @ X5 ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X5: set_list_nat] : ( ord_le6045566169113846134st_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_61_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X5: int] : ( ord_less_eq_int @ X5 @ X5 ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X5: real] : ( ord_less_eq_real @ X5 @ X5 ) ).
% order_refl
thf(fact_64_order__refl,axiom,
! [X5: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_65_dual__order_Orefl,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_66_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A2: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A2: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A2: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_74_Inter__lower,axiom,
! [B: set_se5258582372428582328st_nat,A: set_se7082751667426497518st_nat] :
( ( member4920245652161177295st_nat @ B @ A )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_75_Inter__lower,axiom,
! [B: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_76_Inter__lower,axiom,
! [B: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ B @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_77_Inter__lower,axiom,
! [B: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ B @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_78_Inter__lower,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_79_Inter__lower,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_80_Inter__greatest,axiom,
! [A: set_se7082751667426497518st_nat,C: set_se5258582372428582328st_nat] :
( ! [X2: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X2 @ A )
=> ( ord_le2499698639687704088st_nat @ C @ X2 ) )
=> ( ord_le2499698639687704088st_nat @ C @ ( comple8180622878413557745st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_81_Inter__greatest,axiom,
! [A: set_set_list_nat,C: set_list_nat] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ C @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ C @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_82_Inter__greatest,axiom,
! [A: set_set_set_list_nat,C: set_set_list_nat] :
( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( ord_le1068707526560357548st_nat @ C @ X2 ) )
=> ( ord_le1068707526560357548st_nat @ C @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_83_Inter__greatest,axiom,
! [A: set_se5258582372428582328st_nat,C: set_set_set_list_nat] :
( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ A )
=> ( ord_le7100322305783427298st_nat @ C @ X2 ) )
=> ( ord_le7100322305783427298st_nat @ C @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_84_Inter__greatest,axiom,
! [A: set_set_nat,C: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ C @ X2 ) )
=> ( ord_less_eq_set_nat @ C @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_85_Inter__greatest,axiom,
! [A: set_set_a,C: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ C @ X2 ) )
=> ( ord_less_eq_set_a @ C @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_86_Inter__anti__mono,axiom,
! [B: set_se7082751667426497518st_nat,A: set_se7082751667426497518st_nat] :
( ( ord_le647588107330380878st_nat @ B @ A )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ ( comple8180622878413557745st_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_87_Inter__anti__mono,axiom,
! [B: set_set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_88_Inter__anti__mono,axiom,
! [B: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_89_Inter__anti__mono,axiom,
! [B: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_90_Inter__anti__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_91_Inter__anti__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_92_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_93_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_94_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_95_empty__iff,axiom,
! [C2: list_nat] :
~ ( member_list_nat @ C2 @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_96_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_97_empty__iff,axiom,
! [C2: set_list_nat] :
~ ( member_set_list_nat @ C2 @ bot_bo3886227569956363488st_nat ) ).
% empty_iff
thf(fact_98_empty__iff,axiom,
! [C2: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ C2 @ bot_bo3499706412017099030st_nat ) ).
% empty_iff
thf(fact_99_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_100_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_101_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_102_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_103_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_104_all__not__in__conv,axiom,
! [A: set_set_list_nat] :
( ( ! [X3: set_list_nat] :
~ ( member_set_list_nat @ X3 @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% all_not_in_conv
thf(fact_105_all__not__in__conv,axiom,
! [A: set_set_set_list_nat] :
( ( ! [X3: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ X3 @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% all_not_in_conv
thf(fact_106_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_107_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_108_Collect__empty__eq,axiom,
! [P: set_list_nat > $o] :
( ( ( collect_set_list_nat @ P )
= bot_bo3886227569956363488st_nat )
= ( ! [X3: set_list_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_109_Collect__empty__eq,axiom,
! [P: set_set_list_nat > $o] :
( ( ( collec4691811733418234273st_nat @ P )
= bot_bo3499706412017099030st_nat )
= ( ! [X3: set_set_list_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_110_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_111_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_112_empty__Collect__eq,axiom,
! [P: set_list_nat > $o] :
( ( bot_bo3886227569956363488st_nat
= ( collect_set_list_nat @ P ) )
= ( ! [X3: set_list_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_113_empty__Collect__eq,axiom,
! [P: set_set_list_nat > $o] :
( ( bot_bo3499706412017099030st_nat
= ( collec4691811733418234273st_nat @ P ) )
= ( ! [X3: set_set_list_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_114_image__is__empty,axiom,
! [F: a > nat,A: set_a] :
( ( ( image_a_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_115_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_116_image__is__empty,axiom,
! [F: a > list_nat,A: set_a] :
( ( ( image_a_list_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_117_image__is__empty,axiom,
! [F: list_nat > a,A: set_list_nat] :
( ( ( image_list_nat_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_118_image__is__empty,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( ( image_7976474329151083847st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_119_image__is__empty,axiom,
! [F: set_list_nat > a,A: set_set_list_nat] :
( ( ( image_set_list_nat_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% image_is_empty
thf(fact_120_image__is__empty,axiom,
! [F: a > set_list_nat,A: set_a] :
( ( ( image_a_set_list_nat @ F @ A )
= bot_bo3886227569956363488st_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_121_image__is__empty,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( ( image_6410155499156649085st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% image_is_empty
thf(fact_122_image__is__empty,axiom,
! [F: set_set_list_nat > a,A: set_set_set_list_nat] :
( ( ( image_8692725287357501035_nat_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% image_is_empty
thf(fact_123_image__is__empty,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat] :
( ( ( image_8532145185254316925st_nat @ F @ A )
= bot_bo3886227569956363488st_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_124_empty__is__image,axiom,
! [F: a > nat,A: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_125_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_126_empty__is__image,axiom,
! [F: a > list_nat,A: set_a] :
( ( bot_bot_set_list_nat
= ( image_a_list_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_127_empty__is__image,axiom,
! [F: list_nat > a,A: set_list_nat] :
( ( bot_bot_set_a
= ( image_list_nat_a @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_128_empty__is__image,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_7976474329151083847st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_129_empty__is__image,axiom,
! [F: set_list_nat > a,A: set_set_list_nat] :
( ( bot_bot_set_a
= ( image_set_list_nat_a @ F @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% empty_is_image
thf(fact_130_empty__is__image,axiom,
! [F: a > set_list_nat,A: set_a] :
( ( bot_bo3886227569956363488st_nat
= ( image_a_set_list_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_131_empty__is__image,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_6410155499156649085st_nat @ F @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% empty_is_image
thf(fact_132_empty__is__image,axiom,
! [F: set_set_list_nat > a,A: set_set_set_list_nat] :
( ( bot_bot_set_a
= ( image_8692725287357501035_nat_a @ F @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% empty_is_image
thf(fact_133_empty__is__image,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat] :
( ( bot_bo3886227569956363488st_nat
= ( image_8532145185254316925st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_134_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_135_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_136_image__empty,axiom,
! [F: list_nat > a] :
( ( image_list_nat_a @ F @ bot_bot_set_list_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_137_image__empty,axiom,
! [F: a > list_nat] :
( ( image_a_list_nat @ F @ bot_bot_set_a )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_138_image__empty,axiom,
! [F: list_nat > list_nat] :
( ( image_7976474329151083847st_nat @ F @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_139_image__empty,axiom,
! [F: a > set_list_nat] :
( ( image_a_set_list_nat @ F @ bot_bot_set_a )
= bot_bo3886227569956363488st_nat ) ).
% image_empty
thf(fact_140_image__empty,axiom,
! [F: set_list_nat > a] :
( ( image_set_list_nat_a @ F @ bot_bo3886227569956363488st_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_141_image__empty,axiom,
! [F: list_nat > set_list_nat] :
( ( image_8532145185254316925st_nat @ F @ bot_bot_set_list_nat )
= bot_bo3886227569956363488st_nat ) ).
% image_empty
thf(fact_142_image__empty,axiom,
! [F: a > set_set_list_nat] :
( ( image_6865978040264483717st_nat @ F @ bot_bot_set_a )
= bot_bo3499706412017099030st_nat ) ).
% image_empty
thf(fact_143_image__empty,axiom,
! [F: set_list_nat > list_nat] :
( ( image_6410155499156649085st_nat @ F @ bot_bo3886227569956363488st_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_144_empty__subsetI,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% empty_subsetI
thf(fact_145_empty__subsetI,axiom,
! [A: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ bot_bo3499706412017099030st_nat @ A ) ).
% empty_subsetI
thf(fact_146_empty__subsetI,axiom,
! [A: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ bot_bo3886227569956363488st_nat @ A ) ).
% empty_subsetI
thf(fact_147_empty__subsetI,axiom,
! [A: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ bot_bo1158166727579713100st_nat @ A ) ).
% empty_subsetI
thf(fact_148_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_149_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_150_subset__empty,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_151_subset__empty,axiom,
! [A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ bot_bo3499706412017099030st_nat )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% subset_empty
thf(fact_152_subset__empty,axiom,
! [A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ bot_bo3886227569956363488st_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% subset_empty
thf(fact_153_subset__empty,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ bot_bo1158166727579713100st_nat )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% subset_empty
thf(fact_154_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_155_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_156_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_157_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_158_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_159_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_160_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_161_emptyE,axiom,
! [A2: set_list_nat] :
~ ( member_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ).
% emptyE
thf(fact_162_emptyE,axiom,
! [A2: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ).
% emptyE
thf(fact_163_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_164_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_165_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_166_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_167_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_168_equals0D,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( A = bot_bo3886227569956363488st_nat )
=> ~ ( member_set_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_169_equals0D,axiom,
! [A: set_set_set_list_nat,A2: set_set_list_nat] :
( ( A = bot_bo3499706412017099030st_nat )
=> ~ ( member1029098694177496419st_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_170_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_171_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_172_equals0I,axiom,
! [A: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_173_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y: list_nat] :
~ ( member_list_nat @ Y @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_174_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_175_equals0I,axiom,
! [A: set_set_list_nat] :
( ! [Y: set_list_nat] :
~ ( member_set_list_nat @ Y @ A )
=> ( A = bot_bo3886227569956363488st_nat ) ) ).
% equals0I
thf(fact_176_equals0I,axiom,
! [A: set_set_set_list_nat] :
( ! [Y: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ Y @ A )
=> ( A = bot_bo3499706412017099030st_nat ) ) ).
% equals0I
thf(fact_177_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_178_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_179_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_180_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X3: list_nat] : ( member_list_nat @ X3 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_181_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_182_ex__in__conv,axiom,
! [A: set_set_list_nat] :
( ( ? [X3: set_list_nat] : ( member_set_list_nat @ X3 @ A ) )
= ( A != bot_bo3886227569956363488st_nat ) ) ).
% ex_in_conv
thf(fact_183_ex__in__conv,axiom,
! [A: set_set_set_list_nat] :
( ( ? [X3: set_set_list_nat] : ( member1029098694177496419st_nat @ X3 @ A ) )
= ( A != bot_bo3499706412017099030st_nat ) ) ).
% ex_in_conv
thf(fact_184_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_185_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_186_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ bot_bo3499706412017099030st_nat )
=> ( A2 = bot_bo3499706412017099030st_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_187_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ bot_bo3886227569956363488st_nat )
=> ( A2 = bot_bo3886227569956363488st_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_188_bot_Oextremum__uniqueI,axiom,
! [A2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ bot_bo1158166727579713100st_nat )
=> ( A2 = bot_bo1158166727579713100st_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_189_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_190_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_191_bot_Oextremum__unique,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
= ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_192_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_193_bot_Oextremum__unique,axiom,
! [A2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ bot_bo3499706412017099030st_nat )
= ( A2 = bot_bo3499706412017099030st_nat ) ) ).
% bot.extremum_unique
thf(fact_194_bot_Oextremum__unique,axiom,
! [A2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ bot_bo3886227569956363488st_nat )
= ( A2 = bot_bo3886227569956363488st_nat ) ) ).
% bot.extremum_unique
thf(fact_195_bot_Oextremum__unique,axiom,
! [A2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ bot_bo1158166727579713100st_nat )
= ( A2 = bot_bo1158166727579713100st_nat ) ) ).
% bot.extremum_unique
thf(fact_196_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_197_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_198_bot_Oextremum,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A2 ) ).
% bot.extremum
thf(fact_199_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_200_bot_Oextremum,axiom,
! [A2: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ bot_bo3499706412017099030st_nat @ A2 ) ).
% bot.extremum
thf(fact_201_bot_Oextremum,axiom,
! [A2: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ bot_bo3886227569956363488st_nat @ A2 ) ).
% bot.extremum
thf(fact_202_bot_Oextremum,axiom,
! [A2: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ bot_bo1158166727579713100st_nat @ A2 ) ).
% bot.extremum
thf(fact_203_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_204_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_205_Inf__less__eq,axiom,
! [A: set_se7082751667426497518st_nat,U: set_se5258582372428582328st_nat] :
( ! [V: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ V @ A )
=> ( ord_le2499698639687704088st_nat @ V @ U ) )
=> ( ( A != bot_bo3510429811944912002st_nat )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_206_Inf__less__eq,axiom,
! [A: set_set_list_nat,U: set_list_nat] :
( ! [V: set_list_nat] :
( ( member_set_list_nat @ V @ A )
=> ( ord_le6045566169113846134st_nat @ V @ U ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_207_Inf__less__eq,axiom,
! [A: set_set_set_list_nat,U: set_set_list_nat] :
( ! [V: set_set_list_nat] :
( ( member1029098694177496419st_nat @ V @ A )
=> ( ord_le1068707526560357548st_nat @ V @ U ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_208_Inf__less__eq,axiom,
! [A: set_se5258582372428582328st_nat,U: set_set_set_list_nat] :
( ! [V: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ V @ A )
=> ( ord_le7100322305783427298st_nat @ V @ U ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_209_Inf__less__eq,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V: set_nat] :
( ( member_set_nat @ V @ A )
=> ( ord_less_eq_set_nat @ V @ U ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_210_Inf__less__eq,axiom,
! [A: set_set_a,U: set_a] :
( ! [V: set_a] :
( ( member_set_a @ V @ A )
=> ( ord_less_eq_set_a @ V @ U ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_211_INF__eq__const,axiom,
! [I2: set_nat,F: nat > set_nat,X5: set_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_212_INF__eq__const,axiom,
! [I2: set_a,F: a > set_nat,X5: set_nat] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_213_INF__eq__const,axiom,
! [I2: set_nat,F: nat > set_a,X5: set_a] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_nat_set_a @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_214_INF__eq__const,axiom,
! [I2: set_a,F: a > set_a,X5: set_a] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_215_INF__eq__const,axiom,
! [I2: set_nat,F: nat > set_list_nat,X5: set_list_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_216_INF__eq__const,axiom,
! [I2: set_a,F: a > set_list_nat,X5: set_list_nat] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_217_INF__eq__const,axiom,
! [I2: set_set_nat,F: set_nat > set_nat,X5: set_nat] :
( ( I2 != bot_bot_set_set_nat )
=> ( ! [I: set_nat] :
( ( member_set_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_218_INF__eq__const,axiom,
! [I2: set_set_a,F: set_a > set_nat,X5: set_nat] :
( ( I2 != bot_bot_set_set_a )
=> ( ! [I: set_a] :
( ( member_set_a @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_set_a_set_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_219_INF__eq__const,axiom,
! [I2: set_list_nat,F: list_nat > set_nat,X5: set_nat] :
( ( I2 != bot_bot_set_list_nat )
=> ( ! [I: list_nat] :
( ( member_list_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_220_INF__eq__const,axiom,
! [I2: set_set_nat,F: set_nat > set_a,X5: set_a] :
( ( I2 != bot_bot_set_set_nat )
=> ( ! [I: set_nat] :
( ( member_set_nat @ I @ I2 )
=> ( ( F @ I )
= X5 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_nat_set_a @ F @ I2 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_221_INF__eq__iff,axiom,
! [I2: set_nat,F: nat > set_nat,C2: set_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_222_INF__eq__iff,axiom,
! [I2: set_a,F: a > set_nat,C2: set_nat] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_223_INF__eq__iff,axiom,
! [I2: set_nat,F: nat > set_a,C2: set_a] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_less_eq_set_a @ ( F @ I ) @ C2 ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_nat_set_a @ F @ I2 ) )
= C2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_224_INF__eq__iff,axiom,
! [I2: set_a,F: a > set_a,C2: set_a] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ord_less_eq_set_a @ ( F @ I ) @ C2 ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I2 ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_225_INF__eq__iff,axiom,
! [I2: set_nat,F: nat > set_list_nat,C2: set_list_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_226_INF__eq__iff,axiom,
! [I2: set_a,F: a > set_list_nat,C2: set_list_nat] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_227_INF__eq__iff,axiom,
! [I2: set_set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( I2 != bot_bot_set_set_nat )
=> ( ! [I: set_nat] :
( ( member_set_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_228_INF__eq__iff,axiom,
! [I2: set_set_a,F: set_a > set_nat,C2: set_nat] :
( ( I2 != bot_bot_set_set_a )
=> ( ! [I: set_a] :
( ( member_set_a @ I @ I2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_set_a_set_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_229_INF__eq__iff,axiom,
! [I2: set_list_nat,F: list_nat > set_nat,C2: set_nat] :
( ( I2 != bot_bot_set_list_nat )
=> ( ! [I: list_nat] :
( ( member_list_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ C2 ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ F @ I2 ) )
= C2 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_230_INF__eq__iff,axiom,
! [I2: set_set_nat,F: set_nat > set_a,C2: set_a] :
( ( I2 != bot_bot_set_set_nat )
=> ( ! [I: set_nat] :
( ( member_set_nat @ I @ I2 )
=> ( ord_less_eq_set_a @ ( F @ I ) @ C2 ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_set_nat_set_a @ F @ I2 ) )
= C2 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I2 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_231_order__antisym__conv,axiom,
! [Y2: set_list_nat,X5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y2 @ X5 )
=> ( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_232_order__antisym__conv,axiom,
! [Y2: nat,X5: nat] :
( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_233_order__antisym__conv,axiom,
! [Y2: int,X5: int] :
( ( ord_less_eq_int @ Y2 @ X5 )
=> ( ( ord_less_eq_int @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_234_order__antisym__conv,axiom,
! [Y2: real,X5: real] :
( ( ord_less_eq_real @ Y2 @ X5 )
=> ( ( ord_less_eq_real @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_235_order__antisym__conv,axiom,
! [Y2: set_set_set_list_nat,X5: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ Y2 @ X5 )
=> ( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_236_order__antisym__conv,axiom,
! [Y2: set_set_list_nat,X5: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ Y2 @ X5 )
=> ( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_237_order__antisym__conv,axiom,
! [Y2: set_se5258582372428582328st_nat,X5: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ Y2 @ X5 )
=> ( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_238_order__antisym__conv,axiom,
! [Y2: set_nat,X5: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X5 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_239_order__antisym__conv,axiom,
! [Y2: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X5 )
=> ( ( ord_less_eq_set_a @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_240_linorder__le__cases,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_241_linorder__le__cases,axiom,
! [X5: int,Y2: int] :
( ~ ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_242_linorder__le__cases,axiom,
! [X5: real,Y2: real] :
( ~ ( ord_less_eq_real @ X5 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_243_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_247_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_252_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_258_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_259_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_260_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_262_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_263_linorder__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_264_linorder__linear,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_265_linorder__linear,axiom,
! [X5: real,Y2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_266_order__eq__refl,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( X5 = Y2 )
=> ( ord_le6045566169113846134st_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_267_order__eq__refl,axiom,
! [X5: nat,Y2: nat] :
( ( X5 = Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_268_order__eq__refl,axiom,
! [X5: int,Y2: int] :
( ( X5 = Y2 )
=> ( ord_less_eq_int @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_269_order__eq__refl,axiom,
! [X5: real,Y2: real] :
( ( X5 = Y2 )
=> ( ord_less_eq_real @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_270_order__eq__refl,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( X5 = Y2 )
=> ( ord_le7100322305783427298st_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_271_order__eq__refl,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( X5 = Y2 )
=> ( ord_le1068707526560357548st_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_272_order__eq__refl,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( X5 = Y2 )
=> ( ord_le2499698639687704088st_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_273_order__eq__refl,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( X5 = Y2 )
=> ( ord_less_eq_set_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_274_order__eq__refl,axiom,
! [X5: set_a,Y2: set_a] :
( ( X5 = Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_275_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_276_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_277_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_278_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_279_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_280_order__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_281_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_282_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_283_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_284_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_285_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_286_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_287_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_288_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_289_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_290_order__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_291_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_292_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_293_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_294_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_295_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
& ( ord_le6045566169113846134st_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_296_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_297_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_298_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_299_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A3 @ B3 )
& ( ord_le7100322305783427298st_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_300_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_list_nat,Z: set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A3 @ B3 )
& ( ord_le1068707526560357548st_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_301_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se5258582372428582328st_nat,Z: set_se5258582372428582328st_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_se5258582372428582328st_nat,B3: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A3 @ B3 )
& ( ord_le2499698639687704088st_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_302_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_303_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_304_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_305_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_306_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_307_mem__Collect__eq,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( member_set_list_nat @ A2 @ ( collect_set_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_308_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_309_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_310_mem__Collect__eq,axiom,
! [A2: set_set_list_nat,P: set_set_list_nat > $o] :
( ( member1029098694177496419st_nat @ A2 @ ( collec4691811733418234273st_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_311_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_312_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_313_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_314_Collect__mem__eq,axiom,
! [A: set_set_list_nat] :
( ( collect_set_list_nat
@ ^ [X3: set_list_nat] : ( member_set_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_315_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_316_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_317_Collect__mem__eq,axiom,
! [A: set_set_set_list_nat] :
( ( collec4691811733418234273st_nat
@ ^ [X3: set_set_list_nat] : ( member1029098694177496419st_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_318_antisym,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_319_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_320_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_321_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_322_antisym,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_323_antisym,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_324_antisym,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( ( ord_le2499698639687704088st_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_325_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_326_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_327_dual__order_Otrans,axiom,
! [B2: set_list_nat,A2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ C2 @ B2 )
=> ( ord_le6045566169113846134st_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_328_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_329_dual__order_Otrans,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B2 )
=> ( ord_less_eq_int @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_330_dual__order_Otrans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_331_dual__order_Otrans,axiom,
! [B2: set_set_set_list_nat,A2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ A2 )
=> ( ( ord_le7100322305783427298st_nat @ C2 @ B2 )
=> ( ord_le7100322305783427298st_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_332_dual__order_Otrans,axiom,
! [B2: set_set_list_nat,A2: set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ A2 )
=> ( ( ord_le1068707526560357548st_nat @ C2 @ B2 )
=> ( ord_le1068707526560357548st_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_333_dual__order_Otrans,axiom,
! [B2: set_se5258582372428582328st_nat,A2: set_se5258582372428582328st_nat,C2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B2 @ A2 )
=> ( ( ord_le2499698639687704088st_nat @ C2 @ B2 )
=> ( ord_le2499698639687704088st_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_334_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_335_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_336_dual__order_Oantisym,axiom,
! [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_337_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_338_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_339_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_340_dual__order_Oantisym,axiom,
! [B2: set_set_set_list_nat,A2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ A2 )
=> ( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_341_dual__order_Oantisym,axiom,
! [B2: set_set_list_nat,A2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ A2 )
=> ( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_342_dual__order_Oantisym,axiom,
! [B2: set_se5258582372428582328st_nat,A2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B2 @ A2 )
=> ( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_343_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_344_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_345_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A3 )
& ( ord_le6045566169113846134st_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_346_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_347_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_348_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_349_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B3 @ A3 )
& ( ord_le7100322305783427298st_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_350_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_list_nat,Z: set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B3 @ A3 )
& ( ord_le1068707526560357548st_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_351_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_se5258582372428582328st_nat,Z: set_se5258582372428582328st_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_se5258582372428582328st_nat,B3: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B3 @ A3 )
& ( ord_le2499698639687704088st_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_352_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_355_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_356_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_357_order__trans,axiom,
! [X5: set_list_nat,Y2: set_list_nat,Z2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
=> ( ( ord_le6045566169113846134st_nat @ Y2 @ Z2 )
=> ( ord_le6045566169113846134st_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_358_order__trans,axiom,
! [X5: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_359_order__trans,axiom,
! [X5: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_360_order__trans,axiom,
! [X5: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_361_order__trans,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat,Z2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
=> ( ( ord_le7100322305783427298st_nat @ Y2 @ Z2 )
=> ( ord_le7100322305783427298st_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_362_order__trans,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat,Z2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
=> ( ( ord_le1068707526560357548st_nat @ Y2 @ Z2 )
=> ( ord_le1068707526560357548st_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_363_order__trans,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat,Z2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
=> ( ( ord_le2499698639687704088st_nat @ Y2 @ Z2 )
=> ( ord_le2499698639687704088st_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_364_order__trans,axiom,
! [X5: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_365_order__trans,axiom,
! [X5: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_366_order_Otrans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_367_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_368_order_Otrans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_369_order_Otrans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_370_order_Otrans,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ C2 )
=> ( ord_le7100322305783427298st_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_371_order_Otrans,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ C2 )
=> ( ord_le1068707526560357548st_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_372_order_Otrans,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat,C2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( ( ord_le2499698639687704088st_nat @ B2 @ C2 )
=> ( ord_le2499698639687704088st_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_373_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_374_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_375_order__antisym,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
=> ( ( ord_le6045566169113846134st_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_376_order__antisym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_377_order__antisym,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_378_order__antisym,axiom,
! [X5: real,Y2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_379_order__antisym,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
=> ( ( ord_le7100322305783427298st_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_380_order__antisym,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
=> ( ( ord_le1068707526560357548st_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_381_order__antisym,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
=> ( ( ord_le2499698639687704088st_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_382_order__antisym,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_383_order__antisym,axiom,
! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_384_ord__le__eq__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_385_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_386_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_387_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_388_ord__le__eq__trans,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le7100322305783427298st_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_389_ord__le__eq__trans,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1068707526560357548st_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_390_ord__le__eq__trans,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat,C2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le2499698639687704088st_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_391_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_392_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_393_ord__eq__le__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( A2 = B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_394_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_395_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_396_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_397_ord__eq__le__trans,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( A2 = B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ C2 )
=> ( ord_le7100322305783427298st_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_398_ord__eq__le__trans,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat,C2: set_set_list_nat] :
( ( A2 = B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ C2 )
=> ( ord_le1068707526560357548st_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_399_ord__eq__le__trans,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat,C2: set_se5258582372428582328st_nat] :
( ( A2 = B2 )
=> ( ( ord_le2499698639687704088st_nat @ B2 @ C2 )
=> ( ord_le2499698639687704088st_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_400_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_401_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_402_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
& ( ord_le6045566169113846134st_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_403_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_404_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_405_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_406_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_set_set_list_nat,Y4: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X3 @ Y4 )
& ( ord_le7100322305783427298st_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_407_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_list_nat,Z: set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_set_list_nat,Y4: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X3 @ Y4 )
& ( ord_le1068707526560357548st_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_408_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se5258582372428582328st_nat,Z: set_se5258582372428582328st_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_se5258582372428582328st_nat,Y4: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X3 @ Y4 )
& ( ord_le2499698639687704088st_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_409_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_410_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_411_le__cases3,axiom,
! [X5: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_412_le__cases3,axiom,
! [X5: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X5 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_413_le__cases3,axiom,
! [X5: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X5 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X5 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_414_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_415_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_416_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_417_rev__image__eqI,axiom,
! [X5: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_418_rev__image__eqI,axiom,
! [X5: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_419_rev__image__eqI,axiom,
! [X5: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_420_rev__image__eqI,axiom,
! [X5: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_421_rev__image__eqI,axiom,
! [X5: list_nat,A: set_list_nat,B2: a,F: list_nat > a] :
( ( member_list_nat @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_a @ B2 @ ( image_list_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_422_rev__image__eqI,axiom,
! [X5: list_nat,A: set_list_nat,B2: nat,F: list_nat > nat] :
( ( member_list_nat @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_nat @ B2 @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_423_rev__image__eqI,axiom,
! [X5: a,A: set_a,B2: list_nat,F: a > list_nat] :
( ( member_a @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_list_nat @ B2 @ ( image_a_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_424_rev__image__eqI,axiom,
! [X5: a,A: set_a,B2: set_nat,F: a > set_nat] :
( ( member_a @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_425_rev__image__eqI,axiom,
! [X5: a,A: set_a,B2: set_a,F: a > set_a] :
( ( member_a @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_426_rev__image__eqI,axiom,
! [X5: nat,A: set_nat,B2: list_nat,F: nat > list_nat] :
( ( member_nat @ X5 @ A )
=> ( ( B2
= ( F @ X5 ) )
=> ( member_list_nat @ B2 @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_427_ball__imageD,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,P: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ ( image_8532145185254316925st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: list_nat] :
( ( member_list_nat @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_428_ball__imageD,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,P: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ ( image_920372530894776809st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_429_ball__imageD,axiom,
! [F: a > set_list_nat,A: set_a,P: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ ( image_a_set_list_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_430_ball__imageD,axiom,
! [F: a > set_set_list_nat,A: set_a,P: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ ( image_6865978040264483717st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_431_ball__imageD,axiom,
! [F: a > set_set_set_list_nat,A: set_a,P: set_set_set_list_nat > $o] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ ( image_5164723790120178619st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_432_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_433_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_434_image__cong,axiom,
! [M: set_list_nat,N: set_list_nat,F: list_nat > set_list_nat,G: list_nat > set_list_nat] :
( ( M = N )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_8532145185254316925st_nat @ F @ M )
= ( image_8532145185254316925st_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_435_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_list_nat,G: a > set_list_nat] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_a_set_list_nat @ F @ M )
= ( image_a_set_list_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_436_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_set_list_nat,G: a > set_set_list_nat] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_6865978040264483717st_nat @ F @ M )
= ( image_6865978040264483717st_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_437_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_set_set_list_nat,G: a > set_set_set_list_nat] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_5164723790120178619st_nat @ F @ M )
= ( image_5164723790120178619st_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_438_image__cong,axiom,
! [M: set_a,N: set_a,F: a > nat,G: a > nat] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_a_nat @ F @ M )
= ( image_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_439_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_440_image__cong,axiom,
! [M: set_set_list_nat,N: set_set_list_nat,F: set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat] :
( ( M = N )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_920372530894776809st_nat @ F @ M )
= ( image_920372530894776809st_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_441_bex__imageD,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,P: set_list_nat > $o] :
( ? [X: set_list_nat] :
( ( member_set_list_nat @ X @ ( image_8532145185254316925st_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_442_bex__imageD,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,P: set_set_list_nat > $o] :
( ? [X: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X @ ( image_920372530894776809st_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_443_bex__imageD,axiom,
! [F: a > set_list_nat,A: set_a,P: set_list_nat > $o] :
( ? [X: set_list_nat] :
( ( member_set_list_nat @ X @ ( image_a_set_list_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_444_bex__imageD,axiom,
! [F: a > set_set_list_nat,A: set_a,P: set_set_list_nat > $o] :
( ? [X: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X @ ( image_6865978040264483717st_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_445_bex__imageD,axiom,
! [F: a > set_set_set_list_nat,A: set_a,P: set_set_set_list_nat > $o] :
( ? [X: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X @ ( image_5164723790120178619st_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_446_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_447_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_448_image__iff,axiom,
! [Z2: set_set_set_list_nat,F: a > set_set_set_list_nat,A: set_a] :
( ( member7304678173793621401st_nat @ Z2 @ ( image_5164723790120178619st_nat @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_449_image__iff,axiom,
! [Z2: a,F: a > a,A: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_450_image__iff,axiom,
! [Z2: nat,F: a > nat,A: set_a] :
( ( member_nat @ Z2 @ ( image_a_nat @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_451_image__iff,axiom,
! [Z2: set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( member_set_list_nat @ Z2 @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_452_image__iff,axiom,
! [Z2: set_list_nat,F: a > set_list_nat,A: set_a] :
( ( member_set_list_nat @ Z2 @ ( image_a_set_list_nat @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_453_image__iff,axiom,
! [Z2: set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( member1029098694177496419st_nat @ Z2 @ ( image_920372530894776809st_nat @ F @ A ) )
= ( ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_454_image__iff,axiom,
! [Z2: set_set_list_nat,F: a > set_set_list_nat,A: set_a] :
( ( member1029098694177496419st_nat @ Z2 @ ( image_6865978040264483717st_nat @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_455_imageI,axiom,
! [X5: a,A: set_a,F: a > a] :
( ( member_a @ X5 @ A )
=> ( member_a @ ( F @ X5 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_456_imageI,axiom,
! [X5: a,A: set_a,F: a > nat] :
( ( member_a @ X5 @ A )
=> ( member_nat @ ( F @ X5 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_457_imageI,axiom,
! [X5: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X5 @ A )
=> ( member_a @ ( F @ X5 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_458_imageI,axiom,
! [X5: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X5 @ A )
=> ( member_nat @ ( F @ X5 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_459_imageI,axiom,
! [X5: list_nat,A: set_list_nat,F: list_nat > a] :
( ( member_list_nat @ X5 @ A )
=> ( member_a @ ( F @ X5 ) @ ( image_list_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_460_imageI,axiom,
! [X5: list_nat,A: set_list_nat,F: list_nat > nat] :
( ( member_list_nat @ X5 @ A )
=> ( member_nat @ ( F @ X5 ) @ ( image_list_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_461_imageI,axiom,
! [X5: a,A: set_a,F: a > list_nat] :
( ( member_a @ X5 @ A )
=> ( member_list_nat @ ( F @ X5 ) @ ( image_a_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_462_imageI,axiom,
! [X5: a,A: set_a,F: a > set_nat] :
( ( member_a @ X5 @ A )
=> ( member_set_nat @ ( F @ X5 ) @ ( image_a_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_463_imageI,axiom,
! [X5: a,A: set_a,F: a > set_a] :
( ( member_a @ X5 @ A )
=> ( member_set_a @ ( F @ X5 ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_464_imageI,axiom,
! [X5: nat,A: set_nat,F: nat > list_nat] :
( ( member_nat @ X5 @ A )
=> ( member_list_nat @ ( F @ X5 ) @ ( image_nat_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_465_Sup_OSUP__cong,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat > set_list_nat,D: list_nat > set_list_nat,Sup: set_set_list_nat > set_list_nat] :
( ( A = B )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_8532145185254316925st_nat @ C @ A ) )
= ( Sup @ ( image_8532145185254316925st_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_466_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C: a > set_list_nat,D: a > set_list_nat,Sup: set_set_list_nat > set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_a_set_list_nat @ C @ A ) )
= ( Sup @ ( image_a_set_list_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_467_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C: a > set_set_list_nat,D: a > set_set_list_nat,Sup: set_set_set_list_nat > set_set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_6865978040264483717st_nat @ C @ A ) )
= ( Sup @ ( image_6865978040264483717st_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_468_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C: a > set_set_set_list_nat,D: a > set_set_set_list_nat,Sup: set_se5258582372428582328st_nat > set_set_set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_5164723790120178619st_nat @ C @ A ) )
= ( Sup @ ( image_5164723790120178619st_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_469_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C: a > nat,D: a > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_a_nat @ C @ A ) )
= ( Sup @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_470_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C: a > a,D: a > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_a_a @ C @ A ) )
= ( Sup @ ( image_a_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_471_Sup_OSUP__cong,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,C: set_list_nat > set_set_list_nat,D: set_list_nat > set_set_list_nat,Sup: set_set_set_list_nat > set_set_list_nat] :
( ( A = B )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_920372530894776809st_nat @ C @ A ) )
= ( Sup @ ( image_920372530894776809st_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_472_Inf_OINF__cong,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat > set_list_nat,D: list_nat > set_list_nat,Inf: set_set_list_nat > set_list_nat] :
( ( A = B )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_8532145185254316925st_nat @ C @ A ) )
= ( Inf @ ( image_8532145185254316925st_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_473_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_list_nat,D: a > set_list_nat,Inf: set_set_list_nat > set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_a_set_list_nat @ C @ A ) )
= ( Inf @ ( image_a_set_list_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_474_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_set_list_nat,D: a > set_set_list_nat,Inf: set_set_set_list_nat > set_set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_6865978040264483717st_nat @ C @ A ) )
= ( Inf @ ( image_6865978040264483717st_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_475_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_set_set_list_nat,D: a > set_set_set_list_nat,Inf: set_se5258582372428582328st_nat > set_set_set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_5164723790120178619st_nat @ C @ A ) )
= ( Inf @ ( image_5164723790120178619st_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_476_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C: a > nat,D: a > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_a_nat @ C @ A ) )
= ( Inf @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_477_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C: a > a,D: a > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_a_a @ C @ A ) )
= ( Inf @ ( image_a_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_478_Inf_OINF__cong,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,C: set_list_nat > set_set_list_nat,D: set_list_nat > set_set_list_nat,Inf: set_set_set_list_nat > set_set_list_nat] :
( ( A = B )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_920372530894776809st_nat @ C @ A ) )
= ( Inf @ ( image_920372530894776809st_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_479_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_480_Collect__mono__iff,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ( ord_le7100322305783427298st_nat @ ( collec4691811733418234273st_nat @ P ) @ ( collec4691811733418234273st_nat @ Q ) )
= ( ! [X3: set_set_list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_481_Collect__mono__iff,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ( ord_le1068707526560357548st_nat @ ( collect_set_list_nat @ P ) @ ( collect_set_list_nat @ Q ) )
= ( ! [X3: set_list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_482_Collect__mono__iff,axiom,
! [P: set_set_set_list_nat > $o,Q: set_set_set_list_nat > $o] :
( ( ord_le2499698639687704088st_nat @ ( collec696328324557263319st_nat @ P ) @ ( collec696328324557263319st_nat @ Q ) )
= ( ! [X3: set_set_set_list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_483_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_484_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_485_set__eq__subset,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
& ( ord_le6045566169113846134st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_486_set__eq__subset,axiom,
( ( ^ [Y3: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_set_set_list_nat,B5: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A5 @ B5 )
& ( ord_le7100322305783427298st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_487_set__eq__subset,axiom,
( ( ^ [Y3: set_set_list_nat,Z: set_set_list_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_set_list_nat,B5: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A5 @ B5 )
& ( ord_le1068707526560357548st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_488_set__eq__subset,axiom,
( ( ^ [Y3: set_se5258582372428582328st_nat,Z: set_se5258582372428582328st_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_se5258582372428582328st_nat,B5: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A5 @ B5 )
& ( ord_le2499698639687704088st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_489_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_490_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_491_subset__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_492_subset__trans,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat,C: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B )
=> ( ( ord_le7100322305783427298st_nat @ B @ C )
=> ( ord_le7100322305783427298st_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_493_subset__trans,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,C: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B )
=> ( ( ord_le1068707526560357548st_nat @ B @ C )
=> ( ord_le1068707526560357548st_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_494_subset__trans,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat,C: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ B )
=> ( ( ord_le2499698639687704088st_nat @ B @ C )
=> ( ord_le2499698639687704088st_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_495_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_496_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_497_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_498_Collect__mono,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ ( collec4691811733418234273st_nat @ P ) @ ( collec4691811733418234273st_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_499_Collect__mono,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ ( collect_set_list_nat @ P ) @ ( collect_set_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_500_Collect__mono,axiom,
! [P: set_set_set_list_nat > $o,Q: set_set_set_list_nat > $o] :
( ! [X4: set_set_set_list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le2499698639687704088st_nat @ ( collec696328324557263319st_nat @ P ) @ ( collec696328324557263319st_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_501_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_502_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_503_subset__refl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% subset_refl
thf(fact_504_subset__refl,axiom,
! [A: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ A @ A ) ).
% subset_refl
thf(fact_505_subset__refl,axiom,
! [A: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ A @ A ) ).
% subset_refl
thf(fact_506_subset__refl,axiom,
! [A: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ A @ A ) ).
% subset_refl
thf(fact_507_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_508_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_509_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A5 )
=> ( member_set_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_510_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_511_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T: list_nat] :
( ( member_list_nat @ T @ A5 )
=> ( member_list_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_512_subset__iff,axiom,
( ord_le7100322305783427298st_nat
= ( ^ [A5: set_set_set_list_nat,B5: set_set_set_list_nat] :
! [T: set_set_list_nat] :
( ( member1029098694177496419st_nat @ T @ A5 )
=> ( member1029098694177496419st_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_513_subset__iff,axiom,
( ord_le1068707526560357548st_nat
= ( ^ [A5: set_set_list_nat,B5: set_set_list_nat] :
! [T: set_list_nat] :
( ( member_set_list_nat @ T @ A5 )
=> ( member_set_list_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_514_subset__iff,axiom,
( ord_le2499698639687704088st_nat
= ( ^ [A5: set_se5258582372428582328st_nat,B5: set_se5258582372428582328st_nat] :
! [T: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ T @ A5 )
=> ( member7304678173793621401st_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_515_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_516_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_517_equalityD2,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% equalityD2
thf(fact_518_equalityD2,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat] :
( ( A = B )
=> ( ord_le7100322305783427298st_nat @ B @ A ) ) ).
% equalityD2
thf(fact_519_equalityD2,axiom,
! [A: set_set_list_nat,B: set_set_list_nat] :
( ( A = B )
=> ( ord_le1068707526560357548st_nat @ B @ A ) ) ).
% equalityD2
thf(fact_520_equalityD2,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat] :
( ( A = B )
=> ( ord_le2499698639687704088st_nat @ B @ A ) ) ).
% equalityD2
thf(fact_521_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_522_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_523_equalityD1,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% equalityD1
thf(fact_524_equalityD1,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat] :
( ( A = B )
=> ( ord_le7100322305783427298st_nat @ A @ B ) ) ).
% equalityD1
thf(fact_525_equalityD1,axiom,
! [A: set_set_list_nat,B: set_set_list_nat] :
( ( A = B )
=> ( ord_le1068707526560357548st_nat @ A @ B ) ) ).
% equalityD1
thf(fact_526_equalityD1,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat] :
( ( A = B )
=> ( ord_le2499698639687704088st_nat @ A @ B ) ) ).
% equalityD1
thf(fact_527_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_528_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_529_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_530_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_531_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ A5 )
=> ( member_list_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_532_subset__eq,axiom,
( ord_le7100322305783427298st_nat
= ( ^ [A5: set_set_set_list_nat,B5: set_set_set_list_nat] :
! [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A5 )
=> ( member1029098694177496419st_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_533_subset__eq,axiom,
( ord_le1068707526560357548st_nat
= ( ^ [A5: set_set_list_nat,B5: set_set_list_nat] :
! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A5 )
=> ( member_set_list_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_534_subset__eq,axiom,
( ord_le2499698639687704088st_nat
= ( ^ [A5: set_se5258582372428582328st_nat,B5: set_se5258582372428582328st_nat] :
! [X3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X3 @ A5 )
=> ( member7304678173793621401st_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_535_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_536_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_537_equalityE,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ~ ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ~ ( ord_le6045566169113846134st_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_538_equalityE,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat] :
( ( A = B )
=> ~ ( ( ord_le7100322305783427298st_nat @ A @ B )
=> ~ ( ord_le7100322305783427298st_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_539_equalityE,axiom,
! [A: set_set_list_nat,B: set_set_list_nat] :
( ( A = B )
=> ~ ( ( ord_le1068707526560357548st_nat @ A @ B )
=> ~ ( ord_le1068707526560357548st_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_540_equalityE,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat] :
( ( A = B )
=> ~ ( ( ord_le2499698639687704088st_nat @ A @ B )
=> ~ ( ord_le2499698639687704088st_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_541_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_542_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_543_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_544_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_545_subsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat @ C2 @ A )
=> ( member_list_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_546_subsetD,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B )
=> ( ( member1029098694177496419st_nat @ C2 @ A )
=> ( member1029098694177496419st_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_547_subsetD,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,C2: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B )
=> ( ( member_set_list_nat @ C2 @ A )
=> ( member_set_list_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_548_subsetD,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat,C2: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ A @ B )
=> ( ( member7304678173793621401st_nat @ C2 @ A )
=> ( member7304678173793621401st_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_549_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_550_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_551_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X5: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X5 @ A )
=> ( member_set_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_552_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X5 @ A )
=> ( member_set_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_553_in__mono,axiom,
! [A: set_list_nat,B: set_list_nat,X5: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat @ X5 @ A )
=> ( member_list_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_554_in__mono,axiom,
! [A: set_set_set_list_nat,B: set_set_set_list_nat,X5: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B )
=> ( ( member1029098694177496419st_nat @ X5 @ A )
=> ( member1029098694177496419st_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_555_in__mono,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,X5: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B )
=> ( ( member_set_list_nat @ X5 @ A )
=> ( member_set_list_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_556_in__mono,axiom,
! [A: set_se5258582372428582328st_nat,B: set_se5258582372428582328st_nat,X5: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ A @ B )
=> ( ( member7304678173793621401st_nat @ X5 @ A )
=> ( member7304678173793621401st_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_557_in__mono,axiom,
! [A: set_nat,B: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_558_in__mono,axiom,
! [A: set_a,B: set_a,X5: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X5 @ A )
=> ( member_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_559_InterE,axiom,
! [A: set_nat,C: set_set_set_nat,X6: set_set_nat] :
( ( member_set_nat @ A @ ( comple1065008630642458357et_nat @ C ) )
=> ( ( member_set_set_nat @ X6 @ C )
=> ( member_set_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_560_InterE,axiom,
! [A: set_a,C: set_set_set_a,X6: set_set_a] :
( ( member_set_a @ A @ ( comple9105089376463352645_set_a @ C ) )
=> ( ( member_set_set_a @ X6 @ C )
=> ( member_set_a @ A @ X6 ) ) ) ).
% InterE
thf(fact_561_InterE,axiom,
! [A: list_nat,C: set_set_list_nat,X6: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C ) )
=> ( ( member_set_list_nat @ X6 @ C )
=> ( member_list_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_562_InterE,axiom,
! [A: set_list_nat,C: set_set_set_list_nat,X6: set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C ) )
=> ( ( member1029098694177496419st_nat @ X6 @ C )
=> ( member_set_list_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_563_InterE,axiom,
! [A: set_set_list_nat,C: set_se5258582372428582328st_nat,X6: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A @ ( comple5189992959352112827st_nat @ C ) )
=> ( ( member7304678173793621401st_nat @ X6 @ C )
=> ( member1029098694177496419st_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_564_InterE,axiom,
! [A: nat,C: set_set_nat,X6: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
=> ( ( member_set_nat @ X6 @ C )
=> ( member_nat @ A @ X6 ) ) ) ).
% InterE
thf(fact_565_InterE,axiom,
! [A: a,C: set_set_a,X6: set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C ) )
=> ( ( member_set_a @ X6 @ C )
=> ( member_a @ A @ X6 ) ) ) ).
% InterE
thf(fact_566_InterD,axiom,
! [A: set_nat,C: set_set_set_nat,X6: set_set_nat] :
( ( member_set_nat @ A @ ( comple1065008630642458357et_nat @ C ) )
=> ( ( member_set_set_nat @ X6 @ C )
=> ( member_set_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_567_InterD,axiom,
! [A: set_a,C: set_set_set_a,X6: set_set_a] :
( ( member_set_a @ A @ ( comple9105089376463352645_set_a @ C ) )
=> ( ( member_set_set_a @ X6 @ C )
=> ( member_set_a @ A @ X6 ) ) ) ).
% InterD
thf(fact_568_InterD,axiom,
! [A: list_nat,C: set_set_list_nat,X6: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C ) )
=> ( ( member_set_list_nat @ X6 @ C )
=> ( member_list_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_569_InterD,axiom,
! [A: set_list_nat,C: set_set_set_list_nat,X6: set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C ) )
=> ( ( member1029098694177496419st_nat @ X6 @ C )
=> ( member_set_list_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_570_InterD,axiom,
! [A: set_set_list_nat,C: set_se5258582372428582328st_nat,X6: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A @ ( comple5189992959352112827st_nat @ C ) )
=> ( ( member7304678173793621401st_nat @ X6 @ C )
=> ( member1029098694177496419st_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_571_InterD,axiom,
! [A: nat,C: set_set_nat,X6: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
=> ( ( member_set_nat @ X6 @ C )
=> ( member_nat @ A @ X6 ) ) ) ).
% InterD
thf(fact_572_InterD,axiom,
! [A: a,C: set_set_a,X6: set_a] :
( ( member_a @ A @ ( comple6135023378680113637_set_a @ C ) )
=> ( ( member_set_a @ X6 @ C )
=> ( member_a @ A @ X6 ) ) ) ).
% InterD
thf(fact_573_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_574_subset__image__iff,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_575_subset__image__iff,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_576_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_577_subset__image__iff,axiom,
! [B: set_list_nat,F: nat > list_nat,A: set_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_nat_list_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_list_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_578_subset__image__iff,axiom,
! [B: set_list_nat,F: a > list_nat,A: set_a] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_a_list_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_list_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_579_subset__image__iff,axiom,
! [B: set_nat,F: list_nat > nat,A: set_list_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_list_nat_nat @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B
= ( image_list_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_580_subset__image__iff,axiom,
! [B: set_a,F: list_nat > a,A: set_list_nat] :
( ( ord_less_eq_set_a @ B @ ( image_list_nat_a @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B
= ( image_list_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_581_subset__image__iff,axiom,
! [B: set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_7976474329151083847st_nat @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B
= ( image_7976474329151083847st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_582_subset__image__iff,axiom,
! [B: set_set_list_nat,F: nat > set_list_nat,A: set_nat] :
( ( ord_le1068707526560357548st_nat @ B @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_2883343038133793645st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_583_image__subset__iff,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,B: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_920372530894776809st_nat @ F @ A ) @ B )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( member1029098694177496419st_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_584_image__subset__iff,axiom,
! [F: a > set_set_list_nat,A: set_a,B: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_6865978040264483717st_nat @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member1029098694177496419st_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_585_image__subset__iff,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,B: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_8532145185254316925st_nat @ F @ A ) @ B )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_set_list_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_586_image__subset__iff,axiom,
! [F: a > set_list_nat,A: set_a,B: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_a_set_list_nat @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_set_list_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_587_image__subset__iff,axiom,
! [F: a > set_set_set_list_nat,A: set_a,B: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ ( image_5164723790120178619st_nat @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member7304678173793621401st_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_588_image__subset__iff,axiom,
! [F: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_589_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_590_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_591_subset__imageE,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_592_subset__imageE,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_593_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_594_subset__imageE,axiom,
! [B: set_list_nat,F: nat > list_nat,A: set_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_nat_list_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_list_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_595_subset__imageE,axiom,
! [B: set_list_nat,F: a > list_nat,A: set_a] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_a_list_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_list_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_596_subset__imageE,axiom,
! [B: set_nat,F: list_nat > nat,A: set_list_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_list_nat_nat @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B
!= ( image_list_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_597_subset__imageE,axiom,
! [B: set_a,F: list_nat > a,A: set_list_nat] :
( ( ord_less_eq_set_a @ B @ ( image_list_nat_a @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B
!= ( image_list_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_598_subset__imageE,axiom,
! [B: set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( image_7976474329151083847st_nat @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B
!= ( image_7976474329151083847st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_599_subset__imageE,axiom,
! [B: set_set_list_nat,F: nat > set_list_nat,A: set_nat] :
( ( ord_le1068707526560357548st_nat @ B @ ( image_2883343038133793645st_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_2883343038133793645st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_600_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_601_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_602_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_603_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_604_image__subsetI,axiom,
! [A: set_a,F: a > set_nat,B: set_set_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_605_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_606_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_607_image__subsetI,axiom,
! [A: set_nat,F: nat > set_a,B: set_set_a] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_a @ ( F @ X4 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_608_image__subsetI,axiom,
! [A: set_a,F: a > list_nat,B: set_list_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_list_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6045566169113846134st_nat @ ( image_a_list_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_609_image__subsetI,axiom,
! [A: set_nat,F: nat > list_nat,B: set_list_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_list_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6045566169113846134st_nat @ ( image_nat_list_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_610_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_611_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_612_image__mono,axiom,
! [A: set_a,B: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_613_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_614_image__mono,axiom,
! [A: set_list_nat,B: set_list_nat,F: list_nat > nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F @ A ) @ ( image_list_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_615_image__mono,axiom,
! [A: set_list_nat,B: set_list_nat,F: list_nat > a] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_list_nat_a @ F @ A ) @ ( image_list_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_616_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > list_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ ( image_nat_list_nat @ F @ A ) @ ( image_nat_list_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_617_image__mono,axiom,
! [A: set_a,B: set_a,F: a > list_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le6045566169113846134st_nat @ ( image_a_list_nat @ F @ A ) @ ( image_a_list_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_618_image__mono,axiom,
! [A: set_list_nat,B: set_list_nat,F: list_nat > list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F @ A ) @ ( image_7976474329151083847st_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_619_image__mono,axiom,
! [A: set_set_list_nat,B: set_set_list_nat,F: set_list_nat > nat] :
( ( ord_le1068707526560357548st_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_1475345664228658157at_nat @ F @ A ) @ ( image_1475345664228658157at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_620_Inf__greatest,axiom,
! [A: set_se7082751667426497518st_nat,Z2: set_se5258582372428582328st_nat] :
( ! [X4: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X4 @ A )
=> ( ord_le2499698639687704088st_nat @ Z2 @ X4 ) )
=> ( ord_le2499698639687704088st_nat @ Z2 @ ( comple8180622878413557745st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_621_Inf__greatest,axiom,
! [A: set_set_list_nat,Z2: set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ Z2 @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_622_Inf__greatest,axiom,
! [A: set_set_set_list_nat,Z2: set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ Z2 @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_623_Inf__greatest,axiom,
! [A: set_se5258582372428582328st_nat,Z2: set_set_set_list_nat] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ A )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ Z2 @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_624_Inf__greatest,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ Z2 @ X4 ) )
=> ( ord_less_eq_set_nat @ Z2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_625_Inf__greatest,axiom,
! [A: set_set_a,Z2: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ Z2 @ X4 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_626_le__Inf__iff,axiom,
! [B2: set_se5258582372428582328st_nat,A: set_se7082751667426497518st_nat] :
( ( ord_le2499698639687704088st_nat @ B2 @ ( comple8180622878413557745st_nat @ A ) )
= ( ! [X3: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X3 @ A )
=> ( ord_le2499698639687704088st_nat @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_627_le__Inf__iff,axiom,
! [B2: set_list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( comple184543376406953807st_nat @ A ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_628_le__Inf__iff,axiom,
! [B2: set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( comple8462666950445340293st_nat @ A ) )
= ( ! [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A )
=> ( ord_le1068707526560357548st_nat @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_629_le__Inf__iff,axiom,
! [B2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( comple5189992959352112827st_nat @ A ) )
= ( ! [X3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X3 @ A )
=> ( ord_le7100322305783427298st_nat @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_630_le__Inf__iff,axiom,
! [B2: set_nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( comple7806235888213564991et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_631_le__Inf__iff,axiom,
! [B2: set_a,A: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_632_Inf__lower2,axiom,
! [U: set_se5258582372428582328st_nat,A: set_se7082751667426497518st_nat,V2: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ U @ A )
=> ( ( ord_le2499698639687704088st_nat @ U @ V2 )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_633_Inf__lower2,axiom,
! [U: set_list_nat,A: set_set_list_nat,V2: set_list_nat] :
( ( member_set_list_nat @ U @ A )
=> ( ( ord_le6045566169113846134st_nat @ U @ V2 )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_634_Inf__lower2,axiom,
! [U: set_set_list_nat,A: set_set_set_list_nat,V2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ U @ A )
=> ( ( ord_le1068707526560357548st_nat @ U @ V2 )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_635_Inf__lower2,axiom,
! [U: set_set_set_list_nat,A: set_se5258582372428582328st_nat,V2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ U @ A )
=> ( ( ord_le7100322305783427298st_nat @ U @ V2 )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_636_Inf__lower2,axiom,
! [U: set_nat,A: set_set_nat,V2: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ U @ V2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_637_Inf__lower2,axiom,
! [U: set_a,A: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ U @ V2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_638_Inf__lower,axiom,
! [X5: set_se5258582372428582328st_nat,A: set_se7082751667426497518st_nat] :
( ( member4920245652161177295st_nat @ X5 @ A )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_639_Inf__lower,axiom,
! [X5: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ X5 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_640_Inf__lower,axiom,
! [X5: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ X5 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_641_Inf__lower,axiom,
! [X5: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ X5 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_642_Inf__lower,axiom,
! [X5: set_nat,A: set_set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_643_Inf__lower,axiom,
! [X5: set_a,A: set_set_a] :
( ( member_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_644_Inf__mono,axiom,
! [B: set_se7082751667426497518st_nat,A: set_se7082751667426497518st_nat] :
( ! [B4: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ B4 @ B )
=> ? [X: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X @ A )
& ( ord_le2499698639687704088st_nat @ X @ B4 ) ) )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ ( comple8180622878413557745st_nat @ B ) ) ) ).
% Inf_mono
thf(fact_645_Inf__mono,axiom,
! [B: set_set_list_nat,A: set_set_list_nat] :
( ! [B4: set_list_nat] :
( ( member_set_list_nat @ B4 @ B )
=> ? [X: set_list_nat] :
( ( member_set_list_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ X @ B4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B ) ) ) ).
% Inf_mono
thf(fact_646_Inf__mono,axiom,
! [B: set_set_set_list_nat,A: set_set_set_list_nat] :
( ! [B4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ B4 @ B )
=> ? [X: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X @ A )
& ( ord_le1068707526560357548st_nat @ X @ B4 ) ) )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B ) ) ) ).
% Inf_mono
thf(fact_647_Inf__mono,axiom,
! [B: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ! [B4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ B4 @ B )
=> ? [X: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X @ A )
& ( ord_le7100322305783427298st_nat @ X @ B4 ) ) )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B ) ) ) ).
% Inf_mono
thf(fact_648_Inf__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ! [B4: set_nat] :
( ( member_set_nat @ B4 @ B )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ B4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B ) ) ) ).
% Inf_mono
thf(fact_649_Inf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ! [B4: set_a] :
( ( member_set_a @ B4 @ B )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ B4 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_650_Inf__eqI,axiom,
! [A: set_se7082751667426497518st_nat,X5: set_se5258582372428582328st_nat] :
( ! [I: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ I @ A )
=> ( ord_le2499698639687704088st_nat @ X5 @ I ) )
=> ( ! [Y: set_se5258582372428582328st_nat] :
( ! [I3: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ I3 @ A )
=> ( ord_le2499698639687704088st_nat @ Y @ I3 ) )
=> ( ord_le2499698639687704088st_nat @ Y @ X5 ) )
=> ( ( comple8180622878413557745st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_651_Inf__eqI,axiom,
! [A: set_set_list_nat,X5: set_list_nat] :
( ! [I: set_list_nat] :
( ( member_set_list_nat @ I @ A )
=> ( ord_le6045566169113846134st_nat @ X5 @ I ) )
=> ( ! [Y: set_list_nat] :
( ! [I3: set_list_nat] :
( ( member_set_list_nat @ I3 @ A )
=> ( ord_le6045566169113846134st_nat @ Y @ I3 ) )
=> ( ord_le6045566169113846134st_nat @ Y @ X5 ) )
=> ( ( comple184543376406953807st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_652_Inf__eqI,axiom,
! [A: set_set_set_list_nat,X5: set_set_list_nat] :
( ! [I: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I @ A )
=> ( ord_le1068707526560357548st_nat @ X5 @ I ) )
=> ( ! [Y: set_set_list_nat] :
( ! [I3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I3 @ A )
=> ( ord_le1068707526560357548st_nat @ Y @ I3 ) )
=> ( ord_le1068707526560357548st_nat @ Y @ X5 ) )
=> ( ( comple8462666950445340293st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_653_Inf__eqI,axiom,
! [A: set_se5258582372428582328st_nat,X5: set_set_set_list_nat] :
( ! [I: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I @ A )
=> ( ord_le7100322305783427298st_nat @ X5 @ I ) )
=> ( ! [Y: set_set_set_list_nat] :
( ! [I3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I3 @ A )
=> ( ord_le7100322305783427298st_nat @ Y @ I3 ) )
=> ( ord_le7100322305783427298st_nat @ Y @ X5 ) )
=> ( ( comple5189992959352112827st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_654_Inf__eqI,axiom,
! [A: set_set_nat,X5: set_nat] :
( ! [I: set_nat] :
( ( member_set_nat @ I @ A )
=> ( ord_less_eq_set_nat @ X5 @ I ) )
=> ( ! [Y: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ Y @ I3 ) )
=> ( ord_less_eq_set_nat @ Y @ X5 ) )
=> ( ( comple7806235888213564991et_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_655_Inf__eqI,axiom,
! [A: set_set_a,X5: set_a] :
( ! [I: set_a] :
( ( member_set_a @ I @ A )
=> ( ord_less_eq_set_a @ X5 @ I ) )
=> ( ! [Y: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_set_a @ Y @ I3 ) )
=> ( ord_less_eq_set_a @ Y @ X5 ) )
=> ( ( comple6135023378680113637_set_a @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_656_INF__cong,axiom,
! [A: set_a,B: set_a,C: a > nat,D: a > nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_a_nat @ C @ A ) )
= ( complete_Inf_Inf_nat @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_657_INF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_nat,D: a > set_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_a_set_nat @ C @ A ) )
= ( comple7806235888213564991et_nat @ ( image_a_set_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_658_INF__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C @ A ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_659_INF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_a,D: a > set_a] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ C @ A ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_660_INF__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > set_a,D: nat > set_a] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_nat_set_a @ C @ A ) )
= ( comple6135023378680113637_set_a @ ( image_nat_set_a @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_661_INF__cong,axiom,
! [A: set_a,B: set_a,C: a > set_list_nat,D: a > set_list_nat] :
( ( A = B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ C @ A ) )
= ( comple184543376406953807st_nat @ ( image_a_set_list_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_662_INF__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > set_list_nat,D: nat > set_list_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ C @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_663_INF__cong,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat > set_nat,D: list_nat > set_nat] :
( ( A = B )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ C @ A ) )
= ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_664_INF__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ C @ A ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_665_INF__cong,axiom,
! [A: set_set_a,B: set_set_a,C: set_a > set_nat,D: set_a > set_nat] :
( ( A = B )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_set_a_set_nat @ C @ A ) )
= ( comple7806235888213564991et_nat @ ( image_set_a_set_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_666__092_060sigma_062__max__pointwise__le,axiom,
! [Y5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y5 @ x )
=> ( ( Y5 != bot_bot_set_list_nat )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( max_pointwise @ r @ Y5 ) ) @ n ) ) ) ).
% \<sigma>_max_pointwise_le
thf(fact_667_nlarge,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( max_pointwise @ r @ x ) ) @ n ).
% nlarge
thf(fact_668_cINF__greatest,axiom,
! [A: set_nat,M2: nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_669_cINF__greatest,axiom,
! [A: set_a,M2: nat,F: a > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_670_cINF__greatest,axiom,
! [A: set_nat,M2: int,F: nat > int] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_int @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ M2 @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_671_cINF__greatest,axiom,
! [A: set_a,M2: int,F: a > int] :
( ( A != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_int @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ M2 @ ( complete_Inf_Inf_int @ ( image_a_int @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_672_cINF__greatest,axiom,
! [A: set_nat,M2: real,F: nat > real] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_real @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ M2 @ ( comple4887499456419720421f_real @ ( image_nat_real @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_673_cINF__greatest,axiom,
! [A: set_a,M2: real,F: a > real] :
( ( A != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_real @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ M2 @ ( comple4887499456419720421f_real @ ( image_a_real @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_674_cINF__greatest,axiom,
! [A: set_set_nat,M2: nat,F: set_nat > nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_set_nat_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_675_cINF__greatest,axiom,
! [A: set_set_a,M2: nat,F: set_a > nat] :
( ( A != bot_bot_set_set_a )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_set_a_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_676_cINF__greatest,axiom,
! [A: set_list_nat,M2: nat,F: list_nat > nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_list_nat_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_677_cINF__greatest,axiom,
! [A: set_set_nat,M2: int,F: set_nat > int] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_int @ M2 @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ M2 @ ( complete_Inf_Inf_int @ ( image_set_nat_int @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_678_inj__C,axiom,
inj_on8624761805129053417st_nat @ ( c @ n ) @ x ).
% inj_C
thf(fact_679_max__X,axiom,
! [X7: list_nat] :
( ( member_list_nat @ X7 @ x )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ X7 ) @ ( groups4561878855575611511st_nat @ ( max_pointwise @ r @ x ) ) ) ) ).
% max_X
thf(fact_680_cInf__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X4 ) )
=> ( ! [Y: nat] :
( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ord_less_eq_nat @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_681_cInf__eq__non__empty,axiom,
! [X6: set_int,A2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ A2 @ X4 ) )
=> ( ! [Y: int] :
( ! [X: int] :
( ( member_int @ X @ X6 )
=> ( ord_less_eq_int @ Y @ X ) )
=> ( ord_less_eq_int @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_682_cInf__eq__non__empty,axiom,
! [X6: set_real,A2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ A2 @ X4 ) )
=> ( ! [Y: real] :
( ! [X: real] :
( ( member_real @ X @ X6 )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ord_less_eq_real @ Y @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_683_cInf__eq__non__empty,axiom,
! [X6: set_se7082751667426497518st_nat,A2: set_se5258582372428582328st_nat] :
( ( X6 != bot_bo3510429811944912002st_nat )
=> ( ! [X4: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X4 @ X6 )
=> ( ord_le2499698639687704088st_nat @ A2 @ X4 ) )
=> ( ! [Y: set_se5258582372428582328st_nat] :
( ! [X: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X @ X6 )
=> ( ord_le2499698639687704088st_nat @ Y @ X ) )
=> ( ord_le2499698639687704088st_nat @ Y @ A2 ) )
=> ( ( comple8180622878413557745st_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_684_cInf__eq__non__empty,axiom,
! [X6: set_set_list_nat,A2: set_list_nat] :
( ( X6 != bot_bo3886227569956363488st_nat )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X6 )
=> ( ord_le6045566169113846134st_nat @ A2 @ X4 ) )
=> ( ! [Y: set_list_nat] :
( ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ X6 )
=> ( ord_le6045566169113846134st_nat @ Y @ X ) )
=> ( ord_le6045566169113846134st_nat @ Y @ A2 ) )
=> ( ( comple184543376406953807st_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_685_cInf__eq__non__empty,axiom,
! [X6: set_set_set_list_nat,A2: set_set_list_nat] :
( ( X6 != bot_bo3499706412017099030st_nat )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X6 )
=> ( ord_le1068707526560357548st_nat @ A2 @ X4 ) )
=> ( ! [Y: set_set_list_nat] :
( ! [X: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X @ X6 )
=> ( ord_le1068707526560357548st_nat @ Y @ X ) )
=> ( ord_le1068707526560357548st_nat @ Y @ A2 ) )
=> ( ( comple8462666950445340293st_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_686_cInf__eq__non__empty,axiom,
! [X6: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] :
( ( X6 != bot_bo1158166727579713100st_nat )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X6 )
=> ( ord_le7100322305783427298st_nat @ A2 @ X4 ) )
=> ( ! [Y: set_set_set_list_nat] :
( ! [X: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X @ X6 )
=> ( ord_le7100322305783427298st_nat @ Y @ X ) )
=> ( ord_le7100322305783427298st_nat @ Y @ A2 ) )
=> ( ( comple5189992959352112827st_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_687_cInf__eq__non__empty,axiom,
! [X6: set_set_nat,A2: set_nat] :
( ( X6 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X6 )
=> ( ord_less_eq_set_nat @ A2 @ X4 ) )
=> ( ! [Y: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ X6 )
=> ( ord_less_eq_set_nat @ Y @ X ) )
=> ( ord_less_eq_set_nat @ Y @ A2 ) )
=> ( ( comple7806235888213564991et_nat @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_688_cInf__eq__non__empty,axiom,
! [X6: set_set_a,A2: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ A2 @ X4 ) )
=> ( ! [Y: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( ord_less_eq_set_a @ Y @ A2 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_689_cInf__greatest,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ( ord_less_eq_nat @ Z2 @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_690_cInf__greatest,axiom,
! [X6: set_int,Z2: int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ Z2 @ X4 ) )
=> ( ord_less_eq_int @ Z2 @ ( complete_Inf_Inf_int @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_691_cInf__greatest,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ Z2 @ X4 ) )
=> ( ord_less_eq_real @ Z2 @ ( comple4887499456419720421f_real @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_692_cInf__greatest,axiom,
! [X6: set_se7082751667426497518st_nat,Z2: set_se5258582372428582328st_nat] :
( ( X6 != bot_bo3510429811944912002st_nat )
=> ( ! [X4: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X4 @ X6 )
=> ( ord_le2499698639687704088st_nat @ Z2 @ X4 ) )
=> ( ord_le2499698639687704088st_nat @ Z2 @ ( comple8180622878413557745st_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_693_cInf__greatest,axiom,
! [X6: set_set_list_nat,Z2: set_list_nat] :
( ( X6 != bot_bo3886227569956363488st_nat )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X6 )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ Z2 @ ( comple184543376406953807st_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_694_cInf__greatest,axiom,
! [X6: set_set_set_list_nat,Z2: set_set_list_nat] :
( ( X6 != bot_bo3499706412017099030st_nat )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X6 )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ Z2 @ ( comple8462666950445340293st_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_695_cInf__greatest,axiom,
! [X6: set_se5258582372428582328st_nat,Z2: set_set_set_list_nat] :
( ( X6 != bot_bo1158166727579713100st_nat )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X6 )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ Z2 @ ( comple5189992959352112827st_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_696_cInf__greatest,axiom,
! [X6: set_set_nat,Z2: set_nat] :
( ( X6 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X6 )
=> ( ord_less_eq_set_nat @ Z2 @ X4 ) )
=> ( ord_less_eq_set_nat @ Z2 @ ( comple7806235888213564991et_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_697_cInf__greatest,axiom,
! [X6: set_set_a,Z2: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ Z2 @ X4 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_698_len__X__r,axiom,
! [X5: list_nat] :
( ( member_list_nat @ X5 @ x )
=> ( ( size_size_list_nat @ X5 )
= r ) ) ).
% len_X_r
thf(fact_699_n__ge__X,axiom,
! [X5: list_nat] :
( ( member_list_nat @ X5 @ x )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ X5 ) @ n ) ) ).
% n_ge_X
thf(fact_700__092_060open_062r_A_092_060noteq_062_A0_092_060close_062,axiom,
r != zero_zero_nat ).
% \<open>r \<noteq> 0\<close>
thf(fact_701_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_702_cInf__eq,axiom,
! [X6: set_nat,A2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ A2 @ X4 ) )
=> ( ! [Y: nat] :
( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ord_less_eq_nat @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_703_cInf__eq,axiom,
! [X6: set_int,A2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ A2 @ X4 ) )
=> ( ! [Y: int] :
( ! [X: int] :
( ( member_int @ X @ X6 )
=> ( ord_less_eq_int @ Y @ X ) )
=> ( ord_less_eq_int @ Y @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_704_cInf__eq,axiom,
! [X6: set_real,A2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ A2 @ X4 ) )
=> ( ! [Y: real] :
( ! [X: real] :
( ( member_real @ X @ X6 )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ord_less_eq_real @ Y @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_705_inj__on__Inter,axiom,
! [S: set_set_list_nat,F: list_nat > set_list_nat] :
( ( S != bot_bo3886227569956363488st_nat )
=> ( ! [A6: set_list_nat] :
( ( member_set_list_nat @ A6 @ S )
=> ( inj_on8624761805129053417st_nat @ F @ A6 ) )
=> ( inj_on8624761805129053417st_nat @ F @ ( comple184543376406953807st_nat @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_706_inj__on__Inter,axiom,
! [S: set_set_set_list_nat,F: set_list_nat > set_set_list_nat] :
( ( S != bot_bo3499706412017099030st_nat )
=> ( ! [A6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ A6 @ S )
=> ( inj_on211238342484403029st_nat @ F @ A6 ) )
=> ( inj_on211238342484403029st_nat @ F @ ( comple8462666950445340293st_nat @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_707_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_708_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_709_bot__set__def,axiom,
( bot_bo3886227569956363488st_nat
= ( collect_set_list_nat @ bot_bo7146503454104809893_nat_o ) ) ).
% bot_set_def
thf(fact_710_bot__set__def,axiom,
( bot_bo3499706412017099030st_nat
= ( collec4691811733418234273st_nat @ bot_bo6889798525461962991_nat_o ) ) ).
% bot_set_def
thf(fact_711_Inter__subset,axiom,
! [A: set_se7082751667426497518st_nat,B: set_se5258582372428582328st_nat] :
( ! [X2: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X2 @ A )
=> ( ord_le2499698639687704088st_nat @ X2 @ B ) )
=> ( ( A != bot_bo3510429811944912002st_nat )
=> ( ord_le2499698639687704088st_nat @ ( comple8180622878413557745st_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_712_Inter__subset,axiom,
! [A: set_set_list_nat,B: set_list_nat] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ X2 @ B ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_713_Inter__subset,axiom,
! [A: set_set_set_list_nat,B: set_set_list_nat] :
( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( ord_le1068707526560357548st_nat @ X2 @ B ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_714_Inter__subset,axiom,
! [A: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ A )
=> ( ord_le7100322305783427298st_nat @ X2 @ B ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_715_Inter__subset,axiom,
! [A: set_set_nat,B: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ B ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_716_Inter__subset,axiom,
! [A: set_set_a,B: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ B ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_717_wellorder__InfI,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_718_cInf__eq__minimum,axiom,
! [Z2: nat,X6: set_nat] :
( ( member_nat @ Z2 @ X6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_719_cInf__eq__minimum,axiom,
! [Z2: int,X6: set_int] :
( ( member_int @ Z2 @ X6 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X6 )
=> ( ord_less_eq_int @ Z2 @ X4 ) )
=> ( ( complete_Inf_Inf_int @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_720_cInf__eq__minimum,axiom,
! [Z2: real,X6: set_real] :
( ( member_real @ Z2 @ X6 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ Z2 @ X4 ) )
=> ( ( comple4887499456419720421f_real @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_721_cInf__eq__minimum,axiom,
! [Z2: set_se5258582372428582328st_nat,X6: set_se7082751667426497518st_nat] :
( ( member4920245652161177295st_nat @ Z2 @ X6 )
=> ( ! [X4: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ X4 @ X6 )
=> ( ord_le2499698639687704088st_nat @ Z2 @ X4 ) )
=> ( ( comple8180622878413557745st_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_722_cInf__eq__minimum,axiom,
! [Z2: set_list_nat,X6: set_set_list_nat] :
( ( member_set_list_nat @ Z2 @ X6 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X6 )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ( comple184543376406953807st_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_723_cInf__eq__minimum,axiom,
! [Z2: set_set_list_nat,X6: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ Z2 @ X6 )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X6 )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ( comple8462666950445340293st_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_724_cInf__eq__minimum,axiom,
! [Z2: set_set_set_list_nat,X6: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ Z2 @ X6 )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X6 )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ( comple5189992959352112827st_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_725_cInf__eq__minimum,axiom,
! [Z2: set_nat,X6: set_set_nat] :
( ( member_set_nat @ Z2 @ X6 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X6 )
=> ( ord_less_eq_set_nat @ Z2 @ X4 ) )
=> ( ( comple7806235888213564991et_nat @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_726_cInf__eq__minimum,axiom,
! [Z2: set_a,X6: set_set_a] :
( ( member_set_a @ Z2 @ X6 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ Z2 @ X4 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_727_C__empty__iff,axiom,
! [X7: list_nat] :
( ( ( size_size_list_nat @ X7 )
!= zero_zero_nat )
=> ( ( ( c @ n @ X7 )
= bot_bot_set_list_nat )
= ( ord_less_nat @ n @ ( groups4561878855575611511st_nat @ X7 ) ) ) ) ).
% C_empty_iff
thf(fact_728_inj__on__empty,axiom,
! [F: list_nat > set_list_nat] : ( inj_on8624761805129053417st_nat @ F @ bot_bot_set_list_nat ) ).
% inj_on_empty
thf(fact_729_inj__on__empty,axiom,
! [F: set_list_nat > set_set_list_nat] : ( inj_on211238342484403029st_nat @ F @ bot_bo3886227569956363488st_nat ) ).
% inj_on_empty
thf(fact_730_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_731_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_732_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_733_Csub,axiom,
! [N2: nat,X7: list_nat] : ( ord_le6045566169113846134st_nat @ ( c @ N2 @ X7 ) @ ( length_sum_set @ ( size_size_list_nat @ X7 ) @ N2 ) ) ).
% Csub
thf(fact_734_extend__tuple,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ N2 )
=> ( ( ( size_size_list_nat @ Xs )
!= zero_zero_nat )
=> ~ ! [Ys: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= N2 )
=> ~ ( pointwise_le @ Xs @ Ys ) ) ) ) ).
% extend_tuple
thf(fact_735_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: nat > a] :
( ( image_nat_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_736_inj__on__iff__surj,axiom,
! [A: set_nat,A7: set_a] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F2: nat > a] :
( ( inj_on_nat_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > nat] :
( ( image_a_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_737_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_a] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > a] :
( ( inj_on_a_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > a] :
( ( image_a_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_738_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_list_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > list_nat] :
( ( inj_on_a_list_nat @ F2 @ A )
& ( ord_le6045566169113846134st_nat @ ( image_a_list_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: list_nat > a] :
( ( image_list_nat_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_739_inj__on__iff__surj,axiom,
! [A: set_list_nat,A7: set_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ( ? [F2: list_nat > nat] :
( ( inj_on_list_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: nat > list_nat] :
( ( image_nat_list_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_740_inj__on__iff__surj,axiom,
! [A: set_list_nat,A7: set_a] :
( ( A != bot_bot_set_list_nat )
=> ( ( ? [F2: list_nat > a] :
( ( inj_on_list_nat_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_list_nat_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > list_nat] :
( ( image_a_list_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_741_inj__on__iff__surj,axiom,
! [A: set_list_nat,A7: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ( ? [F2: list_nat > list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ A )
& ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: list_nat > list_nat] :
( ( image_7976474329151083847st_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_742_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_set_list_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > set_list_nat] :
( ( inj_on6747084356327149539st_nat @ F2 @ A )
& ( ord_le1068707526560357548st_nat @ ( image_a_set_list_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: set_list_nat > a] :
( ( image_set_list_nat_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_743_inj__on__iff__surj,axiom,
! [A: set_set_list_nat,A7: set_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( ? [F2: set_list_nat > nat] :
( ( inj_on1516391927397616473at_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_1475345664228658157at_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: nat > set_list_nat] :
( ( image_2883343038133793645st_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_744_inj__on__iff__surj,axiom,
! [A: set_set_list_nat,A7: set_a] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( ? [F2: set_list_nat > a] :
( ( inj_on7310907584571128757_nat_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_set_list_nat_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > set_list_nat] :
( ( image_a_set_list_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_745__092_060open_062max_A1_A_I_092_060sigma_062_A_Imax__pointwise_Ar_AX_J_J_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ ( ord_max_nat @ one_one_nat @ ( groups4561878855575611511st_nat @ ( max_pointwise @ r @ x ) ) ) @ n ).
% \<open>max 1 (\<sigma> (max_pointwise r X)) \<le> n\<close>
thf(fact_746__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_747__092_060open_0621_A_060_Ar_092_060close_062,axiom,
ord_less_nat @ one_one_nat @ r ).
% \<open>1 < r\<close>
thf(fact_748_pointwise__le__refl,axiom,
! [X5: list_nat] : ( pointwise_le @ X5 @ X5 ) ).
% pointwise_le_refl
thf(fact_749_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_750_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_751_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_752_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_753_max__bot,axiom,
! [X5: set_list_nat] :
( ( ord_max_set_list_nat @ bot_bot_set_list_nat @ X5 )
= X5 ) ).
% max_bot
thf(fact_754_max__bot,axiom,
! [X5: set_a] :
( ( ord_max_set_a @ bot_bot_set_a @ X5 )
= X5 ) ).
% max_bot
thf(fact_755_max__bot,axiom,
! [X5: set_set_list_nat] :
( ( ord_ma6513994572189496503st_nat @ bot_bo3886227569956363488st_nat @ X5 )
= X5 ) ).
% max_bot
thf(fact_756_max__bot,axiom,
! [X5: set_set_set_list_nat] :
( ( ord_ma3801742943350824173st_nat @ bot_bo3499706412017099030st_nat @ X5 )
= X5 ) ).
% max_bot
thf(fact_757_max__bot,axiom,
! [X5: nat] :
( ( ord_max_nat @ bot_bot_nat @ X5 )
= X5 ) ).
% max_bot
thf(fact_758_max__bot2,axiom,
! [X5: set_list_nat] :
( ( ord_max_set_list_nat @ X5 @ bot_bot_set_list_nat )
= X5 ) ).
% max_bot2
thf(fact_759_max__bot2,axiom,
! [X5: set_a] :
( ( ord_max_set_a @ X5 @ bot_bot_set_a )
= X5 ) ).
% max_bot2
thf(fact_760_max__bot2,axiom,
! [X5: set_set_list_nat] :
( ( ord_ma6513994572189496503st_nat @ X5 @ bot_bo3886227569956363488st_nat )
= X5 ) ).
% max_bot2
thf(fact_761_max__bot2,axiom,
! [X5: set_set_set_list_nat] :
( ( ord_ma3801742943350824173st_nat @ X5 @ bot_bo3499706412017099030st_nat )
= X5 ) ).
% max_bot2
thf(fact_762_max__bot2,axiom,
! [X5: nat] :
( ( ord_max_nat @ X5 @ bot_bot_nat )
= X5 ) ).
% max_bot2
thf(fact_763_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_764_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_765_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A2 @ B2 ) )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_766_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% max_nat.right_neutral
thf(fact_767_max__0L,axiom,
! [N2: nat] :
( ( ord_max_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% max_0L
thf(fact_768_max__0R,axiom,
! [N2: nat] :
( ( ord_max_nat @ N2 @ zero_zero_nat )
= N2 ) ).
% max_0R
thf(fact_769_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_770__092_060open_062_092_060And_062x_H_O_A_092_060lbrakk_0621_A_060_Ar_059_Ax_H_A_092_060in_062_AX_092_060rbrakk_062_A_092_060Longrightarrow_062_Amin__pointwise_Ar_A_IC_An_Ax_H_J_A_061_Ax_H_092_060close_062,axiom,
! [X7: list_nat] :
( ( ord_less_nat @ one_one_nat @ r )
=> ( ( member_list_nat @ X7 @ x )
=> ( ( min_pointwise @ r @ ( c @ n @ X7 ) )
= X7 ) ) ) ).
% \<open>\<And>x'. \<lbrakk>1 < r; x' \<in> X\<rbrakk> \<Longrightarrow> min_pointwise r (C n x') = x'\<close>
thf(fact_771_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_772_ex__gt__or__lt,axiom,
! [A2: real] :
? [B4: real] :
( ( ord_less_real @ A2 @ B4 )
| ( ord_less_real @ B4 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_773_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_774_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_775_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_776_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_777_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_778_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_779_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_780_one__reorient,axiom,
! [X5: nat] :
( ( one_one_nat = X5 )
= ( X5 = one_one_nat ) ) ).
% one_reorient
thf(fact_781_one__reorient,axiom,
! [X5: int] :
( ( one_one_int = X5 )
= ( X5 = one_one_int ) ) ).
% one_reorient
thf(fact_782_one__reorient,axiom,
! [X5: real] :
( ( one_one_real = X5 )
= ( X5 = one_one_real ) ) ).
% one_reorient
thf(fact_783_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_784_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_785_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_786_linorder__neqE__nat,axiom,
! [X5: nat,Y2: nat] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_neqE_nat
thf(fact_787_pointwise__le__trans,axiom,
! [X5: list_nat,Y2: list_nat,Z2: list_nat] :
( ( pointwise_le @ X5 @ Y2 )
=> ( ( pointwise_le @ Y2 @ Z2 )
=> ( pointwise_le @ X5 @ Z2 ) ) ) ).
% pointwise_le_trans
thf(fact_788_pointwise__le__antisym,axiom,
! [X5: list_nat,Y2: list_nat] :
( ( pointwise_le @ X5 @ Y2 )
=> ( ( pointwise_le @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% pointwise_le_antisym
thf(fact_789_order__less__imp__not__less,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_790_order__less__imp__not__less,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_791_order__less__imp__not__less,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_792_order__less__imp__not__eq2,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_793_order__less__imp__not__eq2,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_794_order__less__imp__not__eq2,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_795_order__less__imp__not__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_796_order__less__imp__not__eq,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_797_order__less__imp__not__eq,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_798_linorder__less__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_799_linorder__less__linear,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_less_int @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_800_linorder__less__linear,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_less_real @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_801_order__less__imp__triv,axiom,
! [X5: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_802_order__less__imp__triv,axiom,
! [X5: int,Y2: int,P: $o] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_803_order__less__imp__triv,axiom,
! [X5: real,Y2: real,P: $o] :
( ( ord_less_real @ X5 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_804_order__less__not__sym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_805_order__less__not__sym,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_806_order__less__not__sym,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_807_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_808_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_809_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_810_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_811_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_812_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_813_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_814_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_815_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_816_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_817_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_818_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_819_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_820_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_821_order__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_822_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_823_order__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_824_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_825_order__less__irrefl,axiom,
! [X5: nat] :
~ ( ord_less_nat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_826_order__less__irrefl,axiom,
! [X5: int] :
~ ( ord_less_int @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_827_order__less__irrefl,axiom,
! [X5: real] :
~ ( ord_less_real @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_828_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_829_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_830_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_831_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_832_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_833_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_834_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_835_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_836_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_837_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_838_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_839_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_840_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_841_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_842_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_843_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_844_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_845_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_846_order__less__trans,axiom,
! [X5: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_847_order__less__trans,axiom,
! [X5: int,Y2: int,Z2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X5 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_848_order__less__trans,axiom,
! [X5: real,Y2: real,Z2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X5 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_849_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_850_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_851_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_852_linorder__neq__iff,axiom,
! [X5: nat,Y2: nat] :
( ( X5 != Y2 )
= ( ( ord_less_nat @ X5 @ Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_853_linorder__neq__iff,axiom,
! [X5: int,Y2: int] :
( ( X5 != Y2 )
= ( ( ord_less_int @ X5 @ Y2 )
| ( ord_less_int @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_854_linorder__neq__iff,axiom,
! [X5: real,Y2: real] :
( ( X5 != Y2 )
= ( ( ord_less_real @ X5 @ Y2 )
| ( ord_less_real @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_855_order__less__asym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_856_order__less__asym,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_857_order__less__asym,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_858_linorder__neqE,axiom,
! [X5: nat,Y2: nat] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_859_linorder__neqE,axiom,
! [X5: int,Y2: int] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_860_linorder__neqE,axiom,
! [X5: real,Y2: real] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_real @ X5 @ Y2 )
=> ( ord_less_real @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_861_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_862_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_863_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_864_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_865_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_866_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_867_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_868_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C2 @ B2 )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_869_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_870_not__less__iff__gr__or__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_871_not__less__iff__gr__or__eq,axiom,
! [X5: int,Y2: int] :
( ( ~ ( ord_less_int @ X5 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_872_not__less__iff__gr__or__eq,axiom,
! [X5: real,Y2: real] :
( ( ~ ( ord_less_real @ X5 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_873_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_874_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_875_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_876_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_877_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_878_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_879_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_880_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_881_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_882_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_883_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_884_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_885_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_886_linorder__cases,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_887_linorder__cases,axiom,
! [X5: int,Y2: int] :
( ~ ( ord_less_int @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_less_int @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_888_linorder__cases,axiom,
! [X5: real,Y2: real] :
( ~ ( ord_less_real @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_less_real @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_889_antisym__conv3,axiom,
! [Y2: nat,X5: nat] :
( ~ ( ord_less_nat @ Y2 @ X5 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_890_antisym__conv3,axiom,
! [Y2: int,X5: int] :
( ~ ( ord_less_int @ Y2 @ X5 )
=> ( ( ~ ( ord_less_int @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_891_antisym__conv3,axiom,
! [Y2: real,X5: real] :
( ~ ( ord_less_real @ Y2 @ X5 )
=> ( ( ~ ( ord_less_real @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_892_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_893_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_894_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_895_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_896_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_897_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C2: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_898_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_899_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_900_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_901_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_902_less__imp__neq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_903_less__imp__neq,axiom,
! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_904_less__imp__neq,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_905_dense,axiom,
! [X5: real,Y2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X5 @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_906_gt__ex,axiom,
! [X5: nat] :
? [X_1: nat] : ( ord_less_nat @ X5 @ X_1 ) ).
% gt_ex
thf(fact_907_gt__ex,axiom,
! [X5: int] :
? [X_1: int] : ( ord_less_int @ X5 @ X_1 ) ).
% gt_ex
thf(fact_908_gt__ex,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% gt_ex
thf(fact_909_lt__ex,axiom,
! [X5: int] :
? [Y: int] : ( ord_less_int @ Y @ X5 ) ).
% lt_ex
thf(fact_910_lt__ex,axiom,
! [X5: real] :
? [Y: real] : ( ord_less_real @ Y @ X5 ) ).
% lt_ex
thf(fact_911_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_912_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_913_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_914_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_915_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_916_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_917_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_918_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J2: nat] :
( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_919_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_920_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_921_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_922_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_923_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_924_linorder__inj__onI_H,axiom,
! [A: set_list_nat,F: list_nat > set_list_nat] :
( ! [I: list_nat,J: list_nat] :
( ( member_list_nat @ I @ A )
=> ( ( member_list_nat @ J @ A )
=> ( ( ord_less_list_nat @ I @ J )
=> ( ( F @ I )
!= ( F @ J ) ) ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_925_max__absorb2,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
=> ( ( ord_max_set_list_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_926_max__absorb2,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_max_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_927_max__absorb2,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_max_int @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_928_max__absorb2,axiom,
! [X5: real,Y2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
=> ( ( ord_max_real @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_929_max__absorb2,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
=> ( ( ord_ma3801742943350824173st_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_930_max__absorb2,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
=> ( ( ord_ma6513994572189496503st_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_931_max__absorb2,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
=> ( ( ord_ma1726305516211428899st_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_932_max__absorb2,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ( ord_max_set_nat @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_933_max__absorb2,axiom,
! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ( ord_max_set_a @ X5 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_934_max__absorb1,axiom,
! [Y2: set_list_nat,X5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y2 @ X5 )
=> ( ( ord_max_set_list_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_935_max__absorb1,axiom,
! [Y2: nat,X5: nat] :
( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ( ord_max_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_936_max__absorb1,axiom,
! [Y2: int,X5: int] :
( ( ord_less_eq_int @ Y2 @ X5 )
=> ( ( ord_max_int @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_937_max__absorb1,axiom,
! [Y2: real,X5: real] :
( ( ord_less_eq_real @ Y2 @ X5 )
=> ( ( ord_max_real @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_938_max__absorb1,axiom,
! [Y2: set_set_set_list_nat,X5: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ Y2 @ X5 )
=> ( ( ord_ma3801742943350824173st_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_939_max__absorb1,axiom,
! [Y2: set_set_list_nat,X5: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ Y2 @ X5 )
=> ( ( ord_ma6513994572189496503st_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_940_max__absorb1,axiom,
! [Y2: set_se5258582372428582328st_nat,X5: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ Y2 @ X5 )
=> ( ( ord_ma1726305516211428899st_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_941_max__absorb1,axiom,
! [Y2: set_nat,X5: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X5 )
=> ( ( ord_max_set_nat @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_942_max__absorb1,axiom,
! [Y2: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X5 )
=> ( ( ord_max_set_a @ X5 @ Y2 )
= X5 ) ) ).
% max_absorb1
thf(fact_943_max__def,axiom,
( ord_max_set_list_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] : ( if_set_list_nat @ ( ord_le6045566169113846134st_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_944_max__def,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_945_max__def,axiom,
( ord_max_int
= ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_946_max__def,axiom,
( ord_max_real
= ( ^ [A3: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_947_max__def,axiom,
( ord_ma3801742943350824173st_nat
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] : ( if_set7539935375068789052st_nat @ ( ord_le7100322305783427298st_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_948_max__def,axiom,
( ord_ma6513994572189496503st_nat
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] : ( if_set_set_list_nat @ ( ord_le1068707526560357548st_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_949_max__def,axiom,
( ord_ma1726305516211428899st_nat
= ( ^ [A3: set_se5258582372428582328st_nat,B3: set_se5258582372428582328st_nat] : ( if_set4742882518001122930st_nat @ ( ord_le2499698639687704088st_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_950_max__def,axiom,
( ord_max_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_951_max__def,axiom,
( ord_max_set_a
= ( ^ [A3: set_a,B3: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_952_order__le__imp__less__or__eq,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
=> ( ( ord_le1190675801316882794st_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_953_order__le__imp__less__or__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_954_order__le__imp__less__or__eq,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_less_int @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_955_order__le__imp__less__or__eq,axiom,
! [X5: real,Y2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
=> ( ( ord_less_real @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_956_order__le__imp__less__or__eq,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
=> ( ( ord_le1759137698939693270st_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_957_order__le__imp__less__or__eq,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
=> ( ( ord_le6760893623726761120st_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_958_order__le__imp__less__or__eq,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
=> ( ( ord_le3416919459018737164st_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_959_order__le__imp__less__or__eq,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ( ord_less_set_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_960_order__le__imp__less__or__eq,axiom,
! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ( ord_less_set_a @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_961_linorder__le__less__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_962_linorder__le__less__linear,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
| ( ord_less_int @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_963_linorder__le__less__linear,axiom,
! [X5: real,Y2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
| ( ord_less_real @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_964_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_965_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_966_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_967_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_968_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_969_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_970_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_971_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_972_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_973_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_974_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_975_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_976_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_977_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_978_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_979_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_980_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_981_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_982_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_983_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_984_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_985_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_986_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_987_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_988_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_989_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_990_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_991_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_992_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_993_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_994_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_995_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_996_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_997_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_998_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_999_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C2: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1000_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1001_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C2: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1002_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1003_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1004_order__less__le__trans,axiom,
! [X5: set_list_nat,Y2: set_list_nat,Z2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X5 @ Y2 )
=> ( ( ord_le6045566169113846134st_nat @ Y2 @ Z2 )
=> ( ord_le1190675801316882794st_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1005_order__less__le__trans,axiom,
! [X5: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1006_order__less__le__trans,axiom,
! [X5: int,Y2: int,Z2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1007_order__less__le__trans,axiom,
! [X5: real,Y2: real,Z2: real] :
( ( ord_less_real @ X5 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1008_order__less__le__trans,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat,Z2: set_set_set_list_nat] :
( ( ord_le1759137698939693270st_nat @ X5 @ Y2 )
=> ( ( ord_le7100322305783427298st_nat @ Y2 @ Z2 )
=> ( ord_le1759137698939693270st_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1009_order__less__le__trans,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat,Z2: set_set_list_nat] :
( ( ord_le6760893623726761120st_nat @ X5 @ Y2 )
=> ( ( ord_le1068707526560357548st_nat @ Y2 @ Z2 )
=> ( ord_le6760893623726761120st_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1010_order__less__le__trans,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat,Z2: set_se5258582372428582328st_nat] :
( ( ord_le3416919459018737164st_nat @ X5 @ Y2 )
=> ( ( ord_le2499698639687704088st_nat @ Y2 @ Z2 )
=> ( ord_le3416919459018737164st_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1011_order__less__le__trans,axiom,
! [X5: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1012_order__less__le__trans,axiom,
! [X5: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X5 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1013_order__le__less__trans,axiom,
! [X5: set_list_nat,Y2: set_list_nat,Z2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ Y2 )
=> ( ( ord_le1190675801316882794st_nat @ Y2 @ Z2 )
=> ( ord_le1190675801316882794st_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1014_order__le__less__trans,axiom,
! [X5: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1015_order__le__less__trans,axiom,
! [X5: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1016_order__le__less__trans,axiom,
! [X5: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X5 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1017_order__le__less__trans,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat,Z2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X5 @ Y2 )
=> ( ( ord_le1759137698939693270st_nat @ Y2 @ Z2 )
=> ( ord_le1759137698939693270st_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1018_order__le__less__trans,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat,Z2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X5 @ Y2 )
=> ( ( ord_le6760893623726761120st_nat @ Y2 @ Z2 )
=> ( ord_le6760893623726761120st_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1019_order__le__less__trans,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat,Z2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ X5 @ Y2 )
=> ( ( ord_le3416919459018737164st_nat @ Y2 @ Z2 )
=> ( ord_le3416919459018737164st_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1020_order__le__less__trans,axiom,
! [X5: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1021_order__le__less__trans,axiom,
! [X5: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X5 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1022_order__neq__le__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( A2 != B2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ord_le1190675801316882794st_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1023_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1024_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1025_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1026_order__neq__le__trans,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( A2 != B2 )
=> ( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( ord_le1759137698939693270st_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1027_order__neq__le__trans,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat] :
( ( A2 != B2 )
=> ( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( ord_le6760893623726761120st_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1028_order__neq__le__trans,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ( A2 != B2 )
=> ( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( ord_le3416919459018737164st_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1029_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1030_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1031_order__le__neq__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le1190675801316882794st_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1032_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1033_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1034_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1035_order__le__neq__trans,axiom,
! [A2: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le1759137698939693270st_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1036_order__le__neq__trans,axiom,
! [A2: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le6760893623726761120st_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1037_order__le__neq__trans,axiom,
! [A2: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le3416919459018737164st_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1038_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1039_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1040_order__less__imp__le,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( ord_le3416919459018737164st_nat @ X5 @ Y2 )
=> ( ord_le2499698639687704088st_nat @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1041_order__less__imp__le,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X5 @ Y2 )
=> ( ord_less_eq_set_nat @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1042_order__less__imp__le,axiom,
! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1043_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1044_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_1045_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1046_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1047_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_1048_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1049_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1050_le__trans,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_1051_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1052_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1053_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1054_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1055_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1056__092_060open_0620_A_092_060le_062_ASUM1_092_060close_062,axiom,
ord_less_eq_int @ zero_zero_int @ sUM1 ).
% \<open>0 \<le> SUM1\<close>
thf(fact_1057__092_060open_062SUM1_A_061_ASUM2_092_060close_062,axiom,
sUM1 = sUM2 ).
% \<open>SUM1 = SUM2\<close>
thf(fact_1058__092_060open_062C_An_A_Ireplicate_Ar_A0_J_A_061_Alength__sum__set_Ar_An_092_060close_062,axiom,
( ( c @ n @ ( replicate_nat @ r @ zero_zero_nat ) )
= ( length_sum_set @ r @ n ) ) ).
% \<open>C n (replicate r 0) = length_sum_set r n\<close>
thf(fact_1059_extend__preserving,axiom,
! [Xs: list_nat,N2: nat,I4: nat] :
( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ~ ! [Ys: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ys )
= N2 )
=> ( ( pointwise_le @ Xs @ Ys )
=> ( ( nth_nat @ Ys @ I4 )
!= ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ) ).
% extend_preserving
thf(fact_1060_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_less @ Xs @ Ys2 )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_1061_Cn0,axiom,
! [N2: nat] :
( ( c @ N2 @ ( replicate_nat @ r @ zero_zero_nat ) )
= ( length_sum_set @ r @ N2 ) ) ).
% Cn0
thf(fact_1062_zero__pointwise__le__iff,axiom,
! [R: nat,X5: list_nat] :
( ( pointwise_le @ ( replicate_nat @ R @ zero_zero_nat ) @ X5 )
= ( ( size_size_list_nat @ X5 )
= R ) ) ).
% zero_pointwise_le_iff
thf(fact_1063_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( pointwise_le @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% pointwise_less_def
thf(fact_1064_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( pointwise_less @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_1065_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( pointwise_le @ X3 @ Y4 )
& ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_size_list_nat @ X3 ) )
& ( ord_less_nat @ ( nth_nat @ X3 @ K3 ) @ ( nth_nat @ Y4 @ K3 ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_1066_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X3 )
= ( size_size_list_nat @ Y4 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ X3 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X3 @ I5 ) @ ( nth_nat @ Y4 @ I5 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_1067_minimal__elementsp_Ocases,axiom,
! [U2: list_nat > $o,A2: list_nat] :
( ( minimal_elementsp @ U2 @ A2 )
=> ~ ( ( U2 @ A2 )
=> ~ ! [Y6: list_nat] :
( ( U2 @ Y6 )
=> ~ ( pointwise_less @ Y6 @ A2 ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_1068_minimal__elementsp_Ointros,axiom,
! [U2: list_nat > $o,X5: list_nat] :
( ( U2 @ X5 )
=> ( ! [Y: list_nat] :
( ( U2 @ Y )
=> ~ ( pointwise_less @ Y @ X5 ) )
=> ( minimal_elementsp @ U2 @ X5 ) ) ) ).
% minimal_elementsp.intros
thf(fact_1069_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U3: list_nat > $o,A3: list_nat] :
? [X3: list_nat] :
( ( A3 = X3 )
& ( U3 @ X3 )
& ! [Y4: list_nat] :
( ( U3 @ Y4 )
=> ~ ( pointwise_less @ Y4 @ X3 ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_1070_minimal__elements_Osimps,axiom,
! [A2: list_nat,U2: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U2 ) )
= ( ? [X3: list_nat] :
( ( A2 = X3 )
& ( member_list_nat @ X3 @ U2 )
& ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U2 )
=> ~ ( pointwise_less @ Y4 @ X3 ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_1071_minimal__elements_Ointros,axiom,
! [X5: list_nat,U2: set_list_nat] :
( ( member_list_nat @ X5 @ U2 )
=> ( ! [Y: list_nat] :
( ( member_list_nat @ Y @ U2 )
=> ~ ( pointwise_less @ Y @ X5 ) )
=> ( member_list_nat @ X5 @ ( minimal_elements @ U2 ) ) ) ) ).
% minimal_elements.intros
thf(fact_1072_minimal__elements_Ocases,axiom,
! [A2: list_nat,U2: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U2 ) )
=> ~ ( ( member_list_nat @ A2 @ U2 )
=> ~ ! [Y6: list_nat] :
( ( member_list_nat @ Y6 @ U2 )
=> ~ ( pointwise_less @ Y6 @ A2 ) ) ) ) ).
% minimal_elements.cases
thf(fact_1073_all__zeroes__replicate,axiom,
! [R: nat] :
( ( length_sum_set @ R @ zero_zero_nat )
= ( insert_list_nat @ ( replicate_nat @ R @ zero_zero_nat ) @ bot_bot_set_list_nat ) ) ).
% all_zeroes_replicate
thf(fact_1074_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1075_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1076_C__eq__1,axiom,
! [X7: list_nat] :
( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ X7 ) @ n )
=> ( ( ( size_size_list_nat @ X7 )
= one_one_nat )
=> ( ( c @ n @ X7 )
= ( insert_list_nat @ ( cons_nat @ n @ nil_nat ) @ bot_bot_set_list_nat ) ) ) ) ).
% C_eq_1
thf(fact_1077_pointwise__le__Nil2,axiom,
! [X5: list_nat] :
( ( pointwise_le @ X5 @ nil_nat )
= ( X5 = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_1078_pointwise__le__Nil,axiom,
! [X5: list_nat] :
( ( pointwise_le @ nil_nat @ X5 )
= ( X5 = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_1079_length__sum__set__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( length_sum_set @ zero_zero_nat @ N2 )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( length_sum_set @ zero_zero_nat @ N2 )
= bot_bot_set_list_nat ) ) ) ).
% length_sum_set_Nil
thf(fact_1080_pointwise__less__Nil2,axiom,
! [X5: list_nat] :
~ ( pointwise_less @ X5 @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_1081_pointwise__less__Nil,axiom,
! [X5: list_nat] :
~ ( pointwise_less @ nil_nat @ X5 ) ).
% pointwise_less_Nil
thf(fact_1082_SUM2__le,axiom,
ord_less_eq_nat @ ( nat2 @ sUM2 ) @ ( finite_card_list_nat @ ( c @ n @ ( replicate_nat @ r @ zero_zero_nat ) ) ) ).
% SUM2_le
thf(fact_1083_minimal__elements__set__tuples__finite,axiom,
! [U2: set_list_nat,R: nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ U2 )
=> ( ( size_size_list_nat @ X4 )
= R ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U2 ) ) ) ).
% minimal_elements_set_tuples_finite
thf(fact_1084_finite__length__sum__set,axiom,
! [R: nat,N2: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R @ N2 ) ) ).
% finite_length_sum_set
thf(fact_1085__092_060open_062finite_AX_092_060close_062,axiom,
finite8100373058378681591st_nat @ x ).
% \<open>finite X\<close>
thf(fact_1086_finC,axiom,
! [N2: nat,X7: list_nat] : ( finite8100373058378681591st_nat @ ( c @ N2 @ X7 ) ) ).
% finC
thf(fact_1087_max__pointwise__ge,axiom,
! [U: list_nat,U2: set_list_nat] :
( ( member_list_nat @ U @ U2 )
=> ( ( finite8100373058378681591st_nat @ U2 )
=> ( pointwise_le @ U @ ( max_pointwise @ ( size_size_list_nat @ U ) @ U2 ) ) ) ) ).
% max_pointwise_ge
thf(fact_1088_min__pointwise__le,axiom,
! [U: list_nat,U2: set_list_nat] :
( ( member_list_nat @ U @ U2 )
=> ( ( finite8100373058378681591st_nat @ U2 )
=> ( pointwise_le @ ( min_pointwise @ ( size_size_list_nat @ U ) @ U2 ) @ U ) ) ) ).
% min_pointwise_le
thf(fact_1089_max__pointwise__le__iff,axiom,
! [U2: set_list_nat,R: nat,X5: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U2 )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X5 )
= R )
=> ( ( pointwise_le @ ( max_pointwise @ R @ U2 ) @ X5 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U2 )
=> ( pointwise_le @ X3 @ X5 ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_1090_max__pointwise__mono,axiom,
! [X9: set_list_nat,X6: set_list_nat,R: nat] :
( ( ord_le6045566169113846134st_nat @ X9 @ X6 )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( X9 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R @ X9 ) @ ( max_pointwise @ R @ X6 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_1091_min__pointwise__ge__iff,axiom,
! [U2: set_list_nat,R: nat,X5: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U2 )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X5 )
= R )
=> ( ( pointwise_le @ X5 @ ( min_pointwise @ R @ U2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U2 )
=> ( pointwise_le @ X5 @ X3 ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_1092_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1093_nat__0__iff,axiom,
! [I4: int] :
( ( ( nat2 @ I4 )
= zero_zero_nat )
= ( ord_less_eq_int @ I4 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1094_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1095_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1096_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1097_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1098_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1099_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1100_nat__mono,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X5 ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_1101_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_1102_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_1103_eq__nat__nat__iff,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z4 ) )
= ( Z2 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1104_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1105_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1106__092_060open_062card_A_IC_An_A_Ireplicate_Ar_A0_J_J_A_061_Acard_A_IU_An_J_A_L_Anat_ASUM2_092_060close_062,axiom,
( ( finite_card_list_nat @ ( c @ n @ ( replicate_nat @ r @ zero_zero_nat ) ) )
= ( plus_plus_nat @ ( finite_card_list_nat @ ( u @ n ) ) @ ( nat2 @ sUM2 ) ) ) ).
% \<open>card (C n (replicate r 0)) = card (U n) + nat SUM2\<close>
thf(fact_1107_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1108_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1109_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1110_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1111_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1112_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1113_trans__less__add2,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1114_trans__less__add1,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1115_add__less__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1116_not__add__less2,axiom,
! [J2: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_1117_not__add__less1,axiom,
! [I4: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ I4 ) ).
% not_add_less1
thf(fact_1118_add__less__mono,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1119_add__lessD1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_1120_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1121_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1122_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1123_trans__le__add2,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1124_trans__le__add1,axiom,
! [I4: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1125_add__le__mono1,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1126_add__le__mono,axiom,
! [I4: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1127_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1128_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1129_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1130_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1131_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1132_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1133_nat__add__max__left,axiom,
! [M2: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N2 ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1134_nat__add__max__right,axiom,
! [M2: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N2 @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1135_less__imp__add__positive,axiom,
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I4 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1136_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1137_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1138_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1139_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1140_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1141_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1142_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1143_int__ge__induct,axiom,
! [K: int,I4: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_ge_induct
thf(fact_1144_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1145_int__gr__induct,axiom,
! [K: int,I4: int,P: int > $o] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_gr_induct
thf(fact_1146_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1147_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1148_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1149_verit__la__generic,axiom,
! [A2: int,X5: int] :
( ( ord_less_eq_int @ A2 @ X5 )
| ( A2 = X5 )
| ( ord_less_eq_int @ X5 @ A2 ) ) ).
% verit_la_generic
thf(fact_1150_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys2 @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_1151_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1152_nat__add__distrib,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z4 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1153_p0,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( p0 @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ ( c @ N2 @ ( replicate_nat @ r @ zero_zero_nat ) ) ) ) ) ) ).
% p0
thf(fact_1154_pf__p0,axiom,
weiers3457258110322917882n_real @ p0 ).
% pf_p0
thf(fact_1155_bounded__Max__nat,axiom,
! [P: nat > $o,X5: nat,M: nat] :
( ( P @ X5 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1156_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1157_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1158_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ord_less_nat @ X4 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1159__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p0_O_A_092_060lbrakk_062real__polynomial__function_Ap0_059_A_092_060And_062n_O_A0_A_060_An_A_092_060Longrightarrow_062_Ap0_A_Ireal_An_J_A_061_Areal_A_Icard_A_IC_An_A_Ireplicate_Ar_A0_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [P0: real > real] :
( ( weiers3457258110322917882n_real @ P0 )
=> ~ ! [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
=> ( ( P0 @ ( semiri5074537144036343181t_real @ N6 ) )
= ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ ( c @ N6 @ ( replicate_nat @ r @ zero_zero_nat ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>p0. \<lbrakk>real_polynomial_function p0; \<And>n. 0 < n \<Longrightarrow> p0 (real n) = real (card (C n (replicate r 0)))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1160_real__polynomial__function__length__sum__set,axiom,
! [R: nat] :
? [P4: real > real] :
( ( weiers3457258110322917882n_real @ P4 )
& ! [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
=> ( ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ ( length_sum_set @ R @ N6 ) ) )
= ( P4 @ ( semiri5074537144036343181t_real @ N6 ) ) ) ) ) ).
% real_polynomial_function_length_sum_set
thf(fact_1161_pf__q,axiom,
weiers3457258110322917882n_real @ q ).
% pf_q
thf(fact_1162_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1163_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1164_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1165_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_1166_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1167_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1168_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1169_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1170_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1171_nat__le__iff,axiom,
! [X5: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X5 ) @ N2 )
= ( ord_less_eq_int @ X5 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1172_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1173_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1174_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1175_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1176_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1177_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1178_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1179_split__nat,axiom,
! [P: nat > $o,I4: int] :
( ( P @ ( nat2 @ I4 ) )
= ( ! [N4: nat] :
( ( I4
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I4 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1180_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1181_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1182_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1183_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M4: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M4 ) ) ) ) ).
% nat_less_real_le
thf(fact_1184_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1185_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1186_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M2 @ N6 )
=> ( ord_less_real @ ( D2 @ N6 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1187_assms_I1_J,axiom,
ord_less_nat @ one_one_nat @ ( finite_card_a @ a2 ) ).
% assms(1)
thf(fact_1188_finA,axiom,
finite_finite_a @ a2 ).
% finA
thf(fact_1189_nonempty,axiom,
a2 != bot_bot_set_a ).
% nonempty
thf(fact_1190_r__def,axiom,
( r
= ( finite_card_a @ a2 ) ) ).
% r_def
thf(fact_1191_idx__less__cardA,axiom,
! [A2: a] :
( ( member_a @ A2 @ a2 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ a2 @ A2 ) @ ( finite_card_a @ a2 ) ) ) ).
% idx_less_cardA
thf(fact_1192_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1193_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1194_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1195_nth__le__list__incr,axiom,
! [I4: nat,X5: list_nat,A2: a] :
( ( ord_less_nat @ I4 @ ( finite_card_a @ a2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X5 @ I4 ) @ ( nth_nat @ ( list_incr @ ( counta3566351752493190365t_on_a @ a2 @ A2 ) @ X5 ) @ I4 ) ) ) ).
% nth_le_list_incr
thf(fact_1196_aA__idx__eq,axiom,
! [A2: a] :
( ( member_a @ A2 @ a2 )
=> ( ( nth_a @ ( aA_a @ a2 ) @ ( counta3566351752493190365t_on_a @ a2 @ A2 ) )
= A2 ) ) ).
% aA_idx_eq
thf(fact_1197_length__list__incr,axiom,
! [I4: nat,X5: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I4 @ X5 ) )
= ( size_size_list_nat @ X5 ) ) ).
% length_list_incr
thf(fact_1198_list__incr__Nil,axiom,
! [I4: nat] :
( ( list_incr @ I4 @ nil_nat )
= nil_nat ) ).
% list_incr_Nil
thf(fact_1199_list__incr__nth__diff,axiom,
! [I4: nat,X5: list_nat,J2: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ X5 ) )
=> ( ( ( I4 = J2 )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J2 @ X5 ) @ I4 ) @ ( nth_nat @ X5 @ I4 ) )
= one_one_nat ) )
& ( ( I4 != J2 )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J2 @ X5 ) @ I4 ) @ ( nth_nat @ X5 @ I4 ) )
= zero_zero_nat ) ) ) ) ).
% list_incr_nth_diff
thf(fact_1200_set__aA,axiom,
( ( set_a2 @ ( aA_a @ a2 ) )
= a2 ) ).
% set_aA
thf(fact_1201_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1202_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1203_diff__diff__cancel,axiom,
! [I4: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1204_diff__diff__left,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1205_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1206_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1207_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1208_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1209_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1210_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1211_nat__minus__add__max,axiom,
! [N2: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M2 ) @ M2 )
= ( ord_max_nat @ N2 @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1212_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1213_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1214_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1215_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1216_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1217_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1218_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1219_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1220_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1221_diff__commute,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1222_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1223_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1224_less__diff__conv,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1225_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1226_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1227_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1228_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1229_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1230_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1231_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1232_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I4 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1233_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1234_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1235_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1236_le__diff__conv,axiom,
! [J2: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1237_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1238_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1239_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1240_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1241_less__diff__conv2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1242_q,axiom,
! [X5: real] :
( ( q @ X5 )
= ( gbinomial_real @ X5 @ ( minus_minus_nat @ r @ one_one_nat ) ) ) ).
% q
thf(fact_1243__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_062real__polynomial__function_Aq_059_A_092_060And_062x_O_Aq_Ax_A_061_Ax_Agchoose_A_Ir_A_N_A1_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Q3: real > real] :
( ( weiers3457258110322917882n_real @ Q3 )
=> ~ ! [X: real] :
( ( Q3 @ X )
= ( gbinomial_real @ X @ ( minus_minus_nat @ r @ one_one_nat ) ) ) ) ).
% \<open>\<And>thesis. (\<And>q. \<lbrakk>real_polynomial_function q; \<And>x. q x = x gchoose (r - 1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1244_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1245_int__le__induct,axiom,
! [I4: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_le_induct
thf(fact_1246_pairwise__minus__cancel,axiom,
! [Z2: list_nat,X5: list_nat,Y2: list_nat] :
( ( pointwise_le @ Z2 @ X5 )
=> ( ( pointwise_le @ Z2 @ Y2 )
=> ( ( ( minus_minus_list_nat @ X5 @ Z2 )
= ( minus_minus_list_nat @ Y2 @ Z2 ) )
=> ( X5 = Y2 ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_1247_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1248_int__less__induct,axiom,
! [I4: int,K: int,P: int > $o] :
( ( ord_less_int @ I4 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_less_induct
thf(fact_1249_int__induct,axiom,
! [P: int > $o,K: int,I4: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% int_induct
thf(fact_1250_sum__list__minus,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys2 @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys2 ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_1251_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1252_nat__diff__distrib,axiom,
! [Z4: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1253_nat__diff__distrib_H,axiom,
! [X5: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( minus_minus_int @ X5 @ Y2 ) )
= ( minus_minus_nat @ ( nat2 @ X5 ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1254_dementum__def,axiom,
( dementum
= ( ^ [Xs2: list_nat] : ( minus_minus_list_nat @ Xs2 @ ( cons_nat @ zero_zero_nat @ Xs2 ) ) ) ) ).
% dementum_def
thf(fact_1255_length__dementum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( dementum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_dementum
thf(fact_1256_dementum__Nil,axiom,
( ( dementum @ nil_nat )
= nil_nat ) ).
% dementum_Nil
thf(fact_1257_zdiff__int__split,axiom,
! [P: int > $o,X5: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X5 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X5 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X5 @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1258_inj__on__imageC,axiom,
inj_on211238342484403029st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ).
% inj_on_imageC
thf(fact_1259_conj__le__cong,axiom,
! [X5: int,X7: int,P: $o,P5: $o] :
( ( X5 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1260_imp__le__cong,axiom,
! [X5: int,X7: int,P: $o,P5: $o] :
( ( X5 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1261__092_060open_062Pow_A_IC_An_A_096_AX_J_A_N_A_123_123_125_125_A_092_060subseteq_062_A_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_092_060close_062,axiom,
ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ ( pow_set_list_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) ) @ ( insert721039030076969596st_nat @ bot_bo3886227569956363488st_nat @ bot_bo3499706412017099030st_nat ) ) @ ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) ).
% \<open>Pow (C n ` X) - {{}} \<subseteq> (`) (C n) ` (Pow X - {{}})\<close>
thf(fact_1262__092_060open_062_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_A_061_APow_A_IC_An_A_096_AX_J_A_N_A_123_123_125_125_092_060close_062,axiom,
( ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) )
= ( minus_7520675481749758683st_nat @ ( pow_set_list_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) ) @ ( insert721039030076969596st_nat @ bot_bo3886227569956363488st_nat @ bot_bo3499706412017099030st_nat ) ) ) ).
% \<open>(`) (C n) ` (Pow X - {{}}) = Pow (C n ` X) - {{}}\<close>
thf(fact_1263_sum__list__augmentum,axiom,
! [Ns: list_nat] :
( ( member_nat @ ( groups4561878855575611511st_nat @ Ns ) @ ( set_nat2 @ ( augmentum @ Ns ) ) )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ns ) ) ) ).
% sum_list_augmentum
thf(fact_1264_augmentum_Osimps_I1_J,axiom,
( ( augmentum @ nil_nat )
= nil_nat ) ).
% augmentum.simps(1)
thf(fact_1265_zero__notin__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) ) ) ).
% zero_notin_augmentum
thf(fact_1266_length__augmentum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( augmentum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_augmentum
% Helper facts (19)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X5: int,Y2: int] :
( ( if_int @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X5: int,Y2: int] :
( ( if_int @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y2: nat] :
( ( if_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y2: nat] :
( ( if_nat @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y2: real] :
( ( if_real @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y2: real] :
( ( if_real @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X5: set_a,Y2: set_a] :
( ( if_set_a @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X5: set_a,Y2: set_a] :
( ( if_set_a @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( if_set_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X5: set_nat,Y2: set_nat] :
( ( if_set_nat @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( if_set_list_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X5: set_list_nat,Y2: set_list_nat] :
( ( if_set_list_nat @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( if_set_set_list_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
! [X5: set_set_list_nat,Y2: set_set_list_nat] :
( ( if_set_set_list_nat @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_T,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( if_set7539935375068789052st_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_T,axiom,
! [X5: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( if_set7539935375068789052st_nat @ $true @ X5 @ Y2 )
= X5 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_T,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( if_set4742882518001122930st_nat @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_T,axiom,
! [X5: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( if_set4742882518001122930st_nat @ $true @ X5 @ Y2 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ y ) ) @ ( c @ n @ ( max_pointwise @ r @ y ) ) ).
%------------------------------------------------------------------------------