TPTP Problem File: SLH0912^1.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_00392_013216__13434766_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1413 ( 368 unt; 141 typ; 0 def)
% Number of atoms : 4311 (1074 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11515 ( 571 ~; 84 |; 273 &;8242 @)
% ( 0 <=>;2345 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 515 ( 515 >; 0 *; 0 +; 0 <<)
% Number of symbols : 131 ( 128 usr; 21 con; 0-3 aty)
% Number of variables : 3704 ( 243 ^;3214 !; 247 ?;3704 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:14:13.459
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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__List__Olist_It__Nat__Onat_J_J,type,
set_list_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__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 (128)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
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__Int__Oint,type,
finite_card_int: set_int > 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_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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_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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__List__Olist_It__Nat__Onat_J,type,
infini2033088105919815547st_nat: set_list_nat > nat > list_nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
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_Olist__incr,type,
list_incr: nat > list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__Int__Oint,type,
khovan4582873290354378414ms_int: set_int > set_int > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__List__Olist_It__Nat__Onat_J,type,
khovan1553326461689229922st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__Nat__Onat,type,
khovan4585363760863428690ms_nat: set_nat > set_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001tf__a,type,
khovanskii_axioms_a: set_a > set_a > $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_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__List__Olist_It__Nat__Onat_J,type,
lattic2817244848751514289st_nat: set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
lattic4275903605611617917x_real: set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Int__Oint,type,
lattic8443796201974363763nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Nat__Onat,type,
lattic8446286672483414039nt_nat: ( int > nat ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic5785867957632790475at_nat: ( list_nat > nat ) > set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Int__Oint,type,
lattic7444442490073309207at_int: ( nat > int ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Int__Oint,type,
lattic6337796949162350289_a_int: ( a > int ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Real__Oreal,type,
lattic7288945864786915537a_real: ( a > real ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Int__Oint,type,
lattic5235898064620869839in_int: set_int > int ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__List__Olist_It__Nat__Onat_J,type,
lattic5191180550204456963st_nat: set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Real__Oreal,type,
lattic2677971596711400399n_real: set_real > real ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic3683530169123051065st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Int__Oint,type,
lattic1091506334969745077in_int: set_int > int ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__List__Olist_It__Nat__Onat_J,type,
lattic6411832703407573737st_nat: set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Real__Oreal,type,
lattic8928443293348198069n_real: set_real > real ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic2169124122975652127st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_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_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
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__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: 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_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__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $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_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__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__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_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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__Nat__Onat_J,type,
is_empty_list_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_a,type,
a3: a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_x,type,
x: list_nat ).
% Relevant facts (1267)
thf(fact_0_idx__less__cardA,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ a2 @ A ) @ ( finite_card_a @ a2 ) ) ) ).
% idx_less_cardA
thf(fact_1_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_2_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ X ) ).
% order_refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_11_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_12_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_13_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_14_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_15_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_16_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_17_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P @ I3 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_18_nonempty,axiom,
a2 != bot_bot_set_a ).
% nonempty
thf(fact_19_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_20_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_21_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_22_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_23_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_24_bot_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
=> ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_25_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_26_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_27_bot_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_28_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_29_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_30_bot_Oextremum,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% bot.extremum
thf(fact_31_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_32_bot_Onot__eq__extremum,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_33_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_34_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_35_bot_Oextremum__strict,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_36_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_37_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_38_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_39_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_40_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_41_order__antisym__conv,axiom,
! [Y: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_42_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_43_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_44_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_45_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_48_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_49_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_51_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_52_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_53_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_54_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_55_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_56_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_57_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_58_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_59_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_60_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_61_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_62_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_63_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_66_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_67_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_68_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_69_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_70_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_71_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_72_order__eq__refl,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( X = Y )
=> ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_73_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_74_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_75_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_76_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_77_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_78_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_79_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_80_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_81_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_82_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_83_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_84_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_85_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_86_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_87_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_88_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_89_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_90_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_91_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_92_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_94_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_95_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_96_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_97_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z2: set_list_nat] : ( Y3 = Z2 ) )
= ( ^ [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
& ( ord_le6045566169113846134st_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_98_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_99_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_100_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_101_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_102_antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_103_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_104_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_105_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_106_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_107_dual__order_Otrans,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le6045566169113846134st_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_108_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_109_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_110_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_111_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_112_dual__order_Oantisym,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_113_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_114_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_115_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_116_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_117_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z2: set_list_nat] : ( Y3 = Z2 ) )
= ( ^ [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
& ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_118_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_119_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_120_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_121_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_122_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_123_order__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_124_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_125_order__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z3 )
=> ( ord_le6045566169113846134st_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_126_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_127_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_128_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_129_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_130_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_131_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_132_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_133_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_134_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_135_order__antisym,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_136_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_137_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_138_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_139_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_140_ord__le__eq__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_141_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_142_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_143_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_144_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_145_ord__eq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_146_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_147_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_148_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_149_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_150_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z2: set_list_nat] : ( Y3 = Z2 ) )
= ( ^ [X4: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X4 @ Y4 )
& ( ord_le6045566169113846134st_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_151_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_152_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_153_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_154_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_155_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_162_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_163_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_164_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_165_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_166_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_167_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_168_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_169_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_170_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_171_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_172_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_173_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_174_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_175_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_176_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_177_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_178_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_179_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_180_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_181_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_182_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_183_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_184_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_185_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_186_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_187_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_188_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_189_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_190_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_191_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_192_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_193_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_194_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_195_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_196_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_197_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_198_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_199_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_200_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_201_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_202_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_203_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_204_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_205_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_206_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_207_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_208_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_209_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_210_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_211_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_212_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_213_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_214_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_215_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_216_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_217_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_218_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_219_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_220_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_221_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_222_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_223_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_224_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_225_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_226_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_227_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_228_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_229_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_230_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_231_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_232_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_233_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_234_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_235_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_236_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_237_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_238_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_239_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_240_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_241_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_242_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_243_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_244_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_245_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_246_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_247_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_248_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_249_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_250_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_251_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_252_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_253_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_254_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_255_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_256_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_257_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_258_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_259_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_260_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_261_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_262_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_263_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_264_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_265_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_266_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_267_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_268_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_269_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_270_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_271_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_272_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_273_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_274_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_275_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_276_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_277_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_278_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_279_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_280_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_281_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_282_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_283_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_284_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_285_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_286_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_287_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_288_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_289_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_290_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_291_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_292_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_293_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_294_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_295_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_296_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_297_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_298_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_299_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_300_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_301_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_302_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_303_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_304_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_305_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_306_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_307_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_308_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_309_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_310_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_311_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_312_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_313_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_314_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_315_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_316_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_317_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_318_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_319_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_320_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_321_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_322_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_323_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_324_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_325_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_326_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_327_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_328_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_329_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_330_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_331_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_332_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_333_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_334_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_335_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_336_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_337_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_338_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_339_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_340_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_341_order__le__imp__less__or__eq,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le1190675801316882794st_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_342_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_343_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_344_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_345_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_346_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_347_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_348_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_349_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_350_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_351_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_352_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_353_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_354_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_355_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_356_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_357_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_358_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_359_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_360_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_361_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_362_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_363_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_364_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_365_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_366_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_367_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_368_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_369_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_370_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_371_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_372_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_373_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_374_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_375_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_376_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_377_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_378_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_379_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_380_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_381_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_382_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_383_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_384_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_385_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_386_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_387_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_388_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_389_order__less__le__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z3 )
=> ( ord_le1190675801316882794st_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_390_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_391_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_392_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_393_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_394_order__le__less__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le1190675801316882794st_nat @ Y @ Z3 )
=> ( ord_le1190675801316882794st_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_395_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_396_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_397_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_398_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_399_order__neq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A != B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_400_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_401_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_402_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_403_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_404_order__le__neq__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_405_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_406_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_407_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_408_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_409_order__less__imp__le,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_410_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_411_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_412_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_413_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_414_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_415_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_416_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_417_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_418_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_419_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_420_order__less__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X4: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_421_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_422_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_423_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_424_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_425_order__le__less,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X4: set_list_nat,Y4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_426_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_427_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_428_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_429_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_430_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_431_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_432_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_433_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_434_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_435_order_Ostrict__implies__order,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_436_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_437_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_438_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_439_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_440_dual__order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
& ~ ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_441_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_442_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_443_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_444_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_445_dual__order_Ostrict__trans2,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_446_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_447_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_448_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_449_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_450_dual__order_Ostrict__trans1,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le1190675801316882794st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_451_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_452_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_453_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_454_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_455_dual__order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_456_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_457_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_458_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_459_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_460_dual__order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_461_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_462_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_463_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_464_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_465_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_466_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_467_order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
& ~ ( ord_le6045566169113846134st_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_468_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_469_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_470_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_471_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_472_order_Ostrict__trans2,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_473_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_474_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_475_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_476_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_477_order_Ostrict__trans1,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_478_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_479_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_480_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_481_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_482_order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_483_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_484_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_485_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_486_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_487_order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_488_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_489_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_490_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_491_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_492_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_493_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_494_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_495_less__le__not__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X4: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X4 @ Y4 )
& ~ ( ord_le6045566169113846134st_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_496_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_497_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_498_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_499_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_500_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_501_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_502_antisym__conv2,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ~ ( ord_le1190675801316882794st_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_503_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_504_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_505_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_506_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_507_antisym__conv1,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ~ ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_508_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_509_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_510_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_511_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_512_nless__le,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ~ ( ord_le1190675801316882794st_nat @ A @ B ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_513_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_514_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_515_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_516_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_517_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_518_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_519_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_520_leD,axiom,
! [Y: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ~ ( ord_le1190675801316882794st_nat @ X @ Y ) ) ).
% leD
thf(fact_521_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_522_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_523_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_524_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_525_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_526_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_527_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_528_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_529_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_530_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_531_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_532_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_533_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_534_subset__empty,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_535_subset__empty,axiom,
! [A4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ bot_bot_set_list_nat )
= ( A4 = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_536_empty__subsetI,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% empty_subsetI
thf(fact_537_empty__subsetI,axiom,
! [A4: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A4 ) ).
% empty_subsetI
thf(fact_538_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_539_all__not__in__conv,axiom,
! [A4: set_int] :
( ( ! [X4: int] :
~ ( member_int @ X4 @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_540_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_541_all__not__in__conv,axiom,
! [A4: set_list_nat] :
( ( ! [X4: list_nat] :
~ ( member_list_nat @ X4 @ A4 ) )
= ( A4 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_542_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_543_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X4: list_nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_544_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_545_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X4: list_nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_546_ex__card,axiom,
! [N2: nat,A4: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ A4 ) )
=> ? [S2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ A4 )
& ( ( finite_card_nat @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_547_ex__card,axiom,
! [N2: nat,A4: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ A4 ) )
=> ? [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ A4 )
& ( ( finite_card_a @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_548_ex__card,axiom,
! [N2: nat,A4: set_list_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ A4 ) )
=> ? [S2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S2 @ A4 )
& ( ( finite_card_list_nat @ S2 )
= N2 ) ) ) ).
% ex_card
thf(fact_549_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_550_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_551_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_552_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_553_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_554_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_555_finA,axiom,
finite_finite_a @ a2 ).
% finA
thf(fact_556_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_557_subsetI,axiom,
! [A4: set_int,B5: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_int @ X3 @ B5 ) )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% subsetI
thf(fact_558_subsetI,axiom,
! [A4: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% subsetI
thf(fact_559_subsetI,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( member_list_nat @ X3 @ B5 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_560_psubsetI,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_a @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_561_psubsetI,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_le1190675801316882794st_nat @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_562_subset__antisym,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_563_subset__antisym,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_564_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_565_in__mono,axiom,
! [A4: set_int,B5: set_int,X: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ X @ A4 )
=> ( member_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_566_in__mono,axiom,
! [A4: set_a,B5: set_a,X: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ X @ A4 )
=> ( member_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_567_in__mono,axiom,
! [A4: set_list_nat,B5: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( member_list_nat @ X @ A4 )
=> ( member_list_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_568_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_569_subsetD,axiom,
! [A4: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_570_subsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_571_subsetD,axiom,
! [A4: set_list_nat,B5: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_572_psubsetE,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_573_psubsetE,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ord_le6045566169113846134st_nat @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_574_equalityE,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ~ ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_575_equalityE,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ~ ( ord_le6045566169113846134st_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_576_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_577_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( member_int @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_578_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_579_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A6 )
=> ( member_list_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_580_equalityD1,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_581_equalityD1,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_582_equalityD2,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ( ord_less_eq_set_a @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_583_equalityD2,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ( ord_le6045566169113846134st_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_584_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_585_psubset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_586_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_587_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_588_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_589_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A6 )
=> ( member_list_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_590_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_591_subset__refl,axiom,
! [A4: set_list_nat] : ( ord_le6045566169113846134st_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_592_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_593_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_594_subset__trans,axiom,
! [A4: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ord_less_eq_set_a @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_595_subset__trans,axiom,
! [A4: set_list_nat,B5: set_list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ C3 )
=> ( ord_le6045566169113846134st_nat @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_596_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_597_set__eq__subset,axiom,
( ( ^ [Y3: set_list_nat,Z2: set_list_nat] : ( Y3 = Z2 ) )
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B6 )
& ( ord_le6045566169113846134st_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_598_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_599_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X4: list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_600_psubset__imp__subset,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_601_psubset__imp__subset,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_602_psubset__subset__trans,axiom,
! [A4: set_a,B5: set_a,C3: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ord_less_set_a @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_603_psubset__subset__trans,axiom,
! [A4: set_list_nat,B5: set_list_nat,C3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ C3 )
=> ( ord_le1190675801316882794st_nat @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_604_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_605_subset__not__subset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B6 )
& ~ ( ord_le6045566169113846134st_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_606_subset__psubset__trans,axiom,
! [A4: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C3 )
=> ( ord_less_set_a @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_607_subset__psubset__trans,axiom,
! [A4: set_list_nat,B5: set_list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_le1190675801316882794st_nat @ B5 @ C3 )
=> ( ord_le1190675801316882794st_nat @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_608_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_609_subset__iff__psubset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_610_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_611_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_612_ex__min__if__finite,axiom,
! [S3: set_list_nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( S3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ S3 )
& ~ ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ S3 )
& ( ord_less_list_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_613_ex__min__if__finite,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S3 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S3 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_614_ex__min__if__finite,axiom,
! [S3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S3 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S3 )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_615_ex__min__if__finite,axiom,
! [S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ( S3 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S3 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S3 )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_616_infinite__growing,axiom,
! [X6: set_list_nat] :
( ( X6 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X6 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ X6 )
& ( ord_less_list_nat @ X3 @ Xa ) ) )
=> ~ ( finite8100373058378681591st_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_617_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_618_infinite__growing,axiom,
! [X6: set_int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X6 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X6 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X6 ) ) ) ).
% infinite_growing
thf(fact_619_infinite__growing,axiom,
! [X6: set_real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X6 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X6 ) ) ) ).
% infinite_growing
thf(fact_620_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_621_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_622_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_623_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_624_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_625_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_626_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_627_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_628_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_629_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_630_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_631_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_632_not__psubset__empty,axiom,
! [A4: set_a] :
~ ( ord_less_set_a @ A4 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_633_not__psubset__empty,axiom,
! [A4: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A4 @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_634_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_635_ex__in__conv,axiom,
! [A4: set_int] :
( ( ? [X4: int] : ( member_int @ X4 @ A4 ) )
= ( A4 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_636_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_637_ex__in__conv,axiom,
! [A4: set_list_nat] :
( ( ? [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
= ( A4 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_638_equals0I,axiom,
! [A4: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_639_equals0I,axiom,
! [A4: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A4 )
=> ( A4 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_640_equals0I,axiom,
! [A4: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_641_equals0I,axiom,
! [A4: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A4 )
=> ( A4 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_642_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_643_equals0D,axiom,
! [A4: set_int,A: int] :
( ( A4 = bot_bot_set_int )
=> ~ ( member_int @ A @ A4 ) ) ).
% equals0D
thf(fact_644_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_645_equals0D,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( A4 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_646_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_647_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_648_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_649_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_650_card__psubset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ( ord_less_set_nat @ A4 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_651_card__psubset,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) )
=> ( ord_less_set_int @ A4 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_652_card__psubset,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_a @ A4 ) @ ( finite_card_a @ B5 ) )
=> ( ord_less_set_a @ A4 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_653_card__psubset,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) )
=> ( ord_le1190675801316882794st_nat @ A4 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_654_card__mono,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% card_mono
thf(fact_655_card__mono,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ).
% card_mono
thf(fact_656_card__mono,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ).
% card_mono
thf(fact_657_card__mono,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ) ).
% card_mono
thf(fact_658_card__seteq,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_seteq
thf(fact_659_card__seteq,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B5 ) @ ( finite_card_int @ A4 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_seteq
thf(fact_660_card__seteq,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B5 ) @ ( finite_card_a @ A4 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_seteq
thf(fact_661_card__seteq,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B5 ) @ ( finite_card_list_nat @ A4 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_seteq
thf(fact_662_exists__subset__between,axiom,
! [A4: set_nat,N2: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C3 )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_663_exists__subset__between,axiom,
! [A4: set_int,N2: nat,C3: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ C3 ) )
=> ( ( ord_less_eq_set_int @ A4 @ C3 )
=> ( ( finite_finite_int @ C3 )
=> ? [B7: set_int] :
( ( ord_less_eq_set_int @ A4 @ B7 )
& ( ord_less_eq_set_int @ B7 @ C3 )
& ( ( finite_card_int @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_664_exists__subset__between,axiom,
! [A4: set_a,N2: nat,C3: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C3 ) )
=> ( ( ord_less_eq_set_a @ A4 @ C3 )
=> ( ( finite_finite_a @ C3 )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A4 @ B7 )
& ( ord_less_eq_set_a @ B7 @ C3 )
& ( ( finite_card_a @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_665_exists__subset__between,axiom,
! [A4: set_list_nat,N2: nat,C3: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ C3 ) )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ C3 )
=> ( ( finite8100373058378681591st_nat @ C3 )
=> ? [B7: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B7 )
& ( ord_le6045566169113846134st_nat @ B7 @ C3 )
& ( ( finite_card_list_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_666_obtain__subset__with__card__n,axiom,
! [N2: nat,S3: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S3 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S3 )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_667_obtain__subset__with__card__n,axiom,
! [N2: nat,S3: set_int] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S3 ) )
=> ~ ! [T3: set_int] :
( ( ord_less_eq_set_int @ T3 @ S3 )
=> ( ( ( finite_card_int @ T3 )
= N2 )
=> ~ ( finite_finite_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_668_obtain__subset__with__card__n,axiom,
! [N2: nat,S3: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S3 ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S3 )
=> ( ( ( finite_card_a @ T3 )
= N2 )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_669_obtain__subset__with__card__n,axiom,
! [N2: nat,S3: set_list_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ S3 ) )
=> ~ ! [T3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ T3 @ S3 )
=> ( ( ( finite_card_list_nat @ T3 )
= N2 )
=> ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_670_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_671_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_int,C3: nat] :
( ! [G: set_int] :
( ( ord_less_eq_set_int @ G @ F2 )
=> ( ( finite_finite_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C3 ) ) )
=> ( ( finite_finite_int @ F2 )
& ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_672_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C3: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C3 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_673_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_nat,C3: nat] :
( ! [G: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G @ F2 )
=> ( ( finite8100373058378681591st_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C3 ) ) )
=> ( ( finite8100373058378681591st_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_674_psubset__card__mono,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_set_a @ A4 @ B5 )
=> ( ord_less_nat @ ( finite_card_a @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_675_psubset__card__mono,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_set_nat @ A4 @ B5 )
=> ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_676_psubset__card__mono,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
=> ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_677_psubset__card__mono,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_set_int @ A4 @ B5 )
=> ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_678_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_int,R: int > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ? [B8: a] :
( ( member_a @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B3: a] :
( ( member_int @ A1 @ A4 )
=> ( ( member_int @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_679_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_a,R: a > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ? [B8: a] :
( ( member_a @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A4 )
=> ( ( member_a @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_680_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B8: a] :
( ( member_a @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_681_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A4: set_int,R: int > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B3: nat] :
( ( member_int @ A1 @ A4 )
=> ( ( member_int @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_682_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A4: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A4 )
=> ( ( member_a @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_683_card__le__if__inj__on__rel,axiom,
! [B5: set_nat,A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_684_card__le__if__inj__on__rel,axiom,
! [B5: set_int,A4: set_int,R: int > int > $o] :
( ( finite_finite_int @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ? [B8: int] :
( ( member_int @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: int,A22: int,B3: int] :
( ( member_int @ A1 @ A4 )
=> ( ( member_int @ A22 @ A4 )
=> ( ( member_int @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_685_card__le__if__inj__on__rel,axiom,
! [B5: set_int,A4: set_a,R: a > int > $o] :
( ( finite_finite_int @ B5 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ? [B8: int] :
( ( member_int @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B3: int] :
( ( member_a @ A1 @ A4 )
=> ( ( member_a @ A22 @ A4 )
=> ( ( member_int @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_686_card__le__if__inj__on__rel,axiom,
! [B5: set_int,A4: set_nat,R: nat > int > $o] :
( ( finite_finite_int @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B8: int] :
( ( member_int @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: int] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_int @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_687_card__le__if__inj__on__rel,axiom,
! [B5: set_a,A4: set_list_nat,R: list_nat > a > $o] :
( ( finite_finite_a @ B5 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ? [B8: a] :
( ( member_a @ B8 @ B5 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B3: a] :
( ( member_list_nat @ A1 @ A4 )
=> ( ( member_list_nat @ A22 @ A4 )
=> ( ( member_a @ B3 @ B5 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_a @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_688_psubsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_689_psubsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_690_psubsetD,axiom,
! [A4: set_list_nat,B5: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
=> ( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_691_psubsetD,axiom,
! [A4: set_int,B5: set_int,C: int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_692_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_693_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_694_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N4 )
=> ( ord_less_nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_695_finite__psubset__induct,axiom,
! [A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ! [B9: set_a] :
( ( ord_less_set_a @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_696_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_697_finite__psubset__induct,axiom,
! [A4: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ! [A7: set_list_nat] :
( ( finite8100373058378681591st_nat @ A7 )
=> ( ! [B9: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_698_finite__psubset__induct,axiom,
! [A4: set_int,P: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ! [B9: set_int] :
( ( ord_less_set_int @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_699_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_700_finite__has__minimal2,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ( ord_less_eq_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A4 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_701_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_702_finite__has__minimal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_703_finite__has__minimal2,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A @ A4 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A4 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_704_finite__has__minimal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A4 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_705_finite__has__minimal2,axiom,
! [A4: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( member_set_list_nat @ A @ A4 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A4 )
& ( ord_le6045566169113846134st_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_706_finite__has__maximal2,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ( ord_less_eq_list_nat @ A @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A4 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_707_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_708_finite__has__maximal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_709_finite__has__maximal2,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A @ A4 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A4 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_710_finite__has__maximal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X3: real] :
( ( member_real @ X3 @ A4 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_711_finite__has__maximal2,axiom,
! [A4: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( member_set_list_nat @ A @ A4 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A4 )
& ( ord_le6045566169113846134st_nat @ A @ X3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_712_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_713_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_714_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_715_finite_OemptyI,axiom,
finite8100373058378681591st_nat @ bot_bot_set_list_nat ).
% finite.emptyI
thf(fact_716_infinite__imp__nonempty,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( S3 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_717_infinite__imp__nonempty,axiom,
! [S3: set_int] :
( ~ ( finite_finite_int @ S3 )
=> ( S3 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_718_infinite__imp__nonempty,axiom,
! [S3: set_a] :
( ~ ( finite_finite_a @ S3 )
=> ( S3 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_719_infinite__imp__nonempty,axiom,
! [S3: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( S3 != bot_bot_set_list_nat ) ) ).
% infinite_imp_nonempty
thf(fact_720_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_721_rev__finite__subset,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_722_rev__finite__subset,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( finite_finite_a @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_723_rev__finite__subset,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_724_infinite__super,axiom,
! [S3: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ T4 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_725_infinite__super,axiom,
! [S3: set_int,T4: set_int] :
( ( ord_less_eq_set_int @ S3 @ T4 )
=> ( ~ ( finite_finite_int @ S3 )
=> ~ ( finite_finite_int @ T4 ) ) ) ).
% infinite_super
thf(fact_726_infinite__super,axiom,
! [S3: set_a,T4: set_a] :
( ( ord_less_eq_set_a @ S3 @ T4 )
=> ( ~ ( finite_finite_a @ S3 )
=> ~ ( finite_finite_a @ T4 ) ) ) ).
% infinite_super
thf(fact_727_infinite__super,axiom,
! [S3: set_list_nat,T4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S3 @ T4 )
=> ( ~ ( finite8100373058378681591st_nat @ S3 )
=> ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_728_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_729_finite__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_subset
thf(fact_730_finite__subset,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( finite_finite_a @ B5 )
=> ( finite_finite_a @ A4 ) ) ) ).
% finite_subset
thf(fact_731_finite__subset,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( finite8100373058378681591st_nat @ B5 )
=> ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_732_finite__has__maximal,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A4 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_733_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_734_finite__has__maximal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_735_finite__has__maximal,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A4 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_736_finite__has__maximal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_737_finite__has__maximal,axiom,
! [A4: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A4 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_738_finite__has__minimal,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A4 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_739_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_740_finite__has__minimal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_741_finite__has__minimal,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A4 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_742_finite__has__minimal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_743_finite__has__minimal,axiom,
! [A4: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A4 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_744_card__subset__eq,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B5 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_745_card__subset__eq,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ( finite_card_int @ A4 )
= ( finite_card_int @ B5 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_746_card__subset__eq,axiom,
! [B5: set_a,A4: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ( finite_card_a @ A4 )
= ( finite_card_a @ B5 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_747_card__subset__eq,axiom,
! [B5: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ( finite_card_list_nat @ A4 )
= ( finite_card_list_nat @ B5 ) )
=> ( A4 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_748_infinite__arbitrarily__large,axiom,
! [A4: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_749_infinite__arbitrarily__large,axiom,
! [A4: set_int,N2: nat] :
( ~ ( finite_finite_int @ A4 )
=> ? [B7: set_int] :
( ( finite_finite_int @ B7 )
& ( ( finite_card_int @ B7 )
= N2 )
& ( ord_less_eq_set_int @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_750_infinite__arbitrarily__large,axiom,
! [A4: set_a,N2: nat] :
( ~ ( finite_finite_a @ A4 )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N2 )
& ( ord_less_eq_set_a @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_751_infinite__arbitrarily__large,axiom,
! [A4: set_list_nat,N2: nat] :
( ~ ( finite8100373058378681591st_nat @ A4 )
=> ? [B7: set_list_nat] :
( ( finite8100373058378681591st_nat @ B7 )
& ( ( finite_card_list_nat @ B7 )
= N2 )
& ( ord_le6045566169113846134st_nat @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_752_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_753_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_754_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_755_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X4: list_nat] : ( member_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_756_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_757_Collect__empty__eq__bot,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( P = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_758_Khovanskii__axioms_Ointro,axiom,
! [A4: set_nat,G2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ G2 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( khovan4585363760863428690ms_nat @ G2 @ A4 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_759_Khovanskii__axioms_Ointro,axiom,
! [A4: set_int,G2: set_int] :
( ( ord_less_eq_set_int @ A4 @ G2 )
=> ( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( khovan4582873290354378414ms_int @ G2 @ A4 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_760_Khovanskii__axioms_Ointro,axiom,
! [A4: set_a,G2: set_a] :
( ( ord_less_eq_set_a @ A4 @ G2 )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( khovanskii_axioms_a @ G2 @ A4 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_761_Khovanskii__axioms_Ointro,axiom,
! [A4: set_list_nat,G2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ G2 )
=> ( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( khovan1553326461689229922st_nat @ G2 @ A4 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_762_Khovanskii__axioms__def,axiom,
( khovan4585363760863428690ms_nat
= ( ^ [G3: set_nat,A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ G3 )
& ( finite_finite_nat @ A6 )
& ( A6 != bot_bot_set_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_763_Khovanskii__axioms__def,axiom,
( khovan4582873290354378414ms_int
= ( ^ [G3: set_int,A6: set_int] :
( ( ord_less_eq_set_int @ A6 @ G3 )
& ( finite_finite_int @ A6 )
& ( A6 != bot_bot_set_int ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_764_Khovanskii__axioms__def,axiom,
( khovanskii_axioms_a
= ( ^ [G3: set_a,A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ G3 )
& ( finite_finite_a @ A6 )
& ( A6 != bot_bot_set_a ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_765_Khovanskii__axioms__def,axiom,
( khovan1553326461689229922st_nat
= ( ^ [G3: set_list_nat,A6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ G3 )
& ( finite8100373058378681591st_nat @ A6 )
& ( A6 != bot_bot_set_list_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_766_arg__min__if__finite_I2_J,axiom,
! [S3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ~ ? [X2: nat] :
( ( member_nat @ X2 @ S3 )
& ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_767_arg__min__if__finite_I2_J,axiom,
! [S3: set_int,F: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ~ ? [X2: int] :
( ( member_int @ X2 @ S3 )
& ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_768_arg__min__if__finite_I2_J,axiom,
! [S3: set_a,F: a > nat] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ~ ? [X2: a] :
( ( member_a @ X2 @ S3 )
& ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_769_arg__min__if__finite_I2_J,axiom,
! [S3: set_nat,F: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ~ ? [X2: nat] :
( ( member_nat @ X2 @ S3 )
& ( ord_less_int @ ( F @ X2 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_770_arg__min__if__finite_I2_J,axiom,
! [S3: set_int,F: int > int] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ~ ? [X2: int] :
( ( member_int @ X2 @ S3 )
& ( ord_less_int @ ( F @ X2 ) @ ( F @ ( lattic8443796201974363763nt_int @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_771_arg__min__if__finite_I2_J,axiom,
! [S3: set_a,F: a > int] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ~ ? [X2: a] :
( ( member_a @ X2 @ S3 )
& ( ord_less_int @ ( F @ X2 ) @ ( F @ ( lattic6337796949162350289_a_int @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_772_arg__min__if__finite_I2_J,axiom,
! [S3: set_nat,F: nat > real] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ~ ? [X2: nat] :
( ( member_nat @ X2 @ S3 )
& ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_773_arg__min__if__finite_I2_J,axiom,
! [S3: set_int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ~ ? [X2: int] :
( ( member_int @ X2 @ S3 )
& ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_774_arg__min__if__finite_I2_J,axiom,
! [S3: set_a,F: a > real] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ~ ? [X2: a] :
( ( member_a @ X2 @ S3 )
& ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic7288945864786915537a_real @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_775_arg__min__if__finite_I2_J,axiom,
! [S3: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( S3 != bot_bot_set_list_nat )
=> ~ ? [X2: list_nat] :
( ( member_list_nat @ X2 @ S3 )
& ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic5785867957632790475at_nat @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_776_arg__min__least,axiom,
! [S3: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_777_arg__min__least,axiom,
! [S3: set_int,Y: int,F: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ( ( member_int @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_778_arg__min__least,axiom,
! [S3: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ( ( member_a @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_779_arg__min__least,axiom,
! [S3: set_nat,Y: nat,F: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S3 )
=> ( ord_less_eq_int @ ( F @ ( lattic7444442490073309207at_int @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_780_arg__min__least,axiom,
! [S3: set_int,Y: int,F: int > int] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ( ( member_int @ Y @ S3 )
=> ( ord_less_eq_int @ ( F @ ( lattic8443796201974363763nt_int @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_781_arg__min__least,axiom,
! [S3: set_a,Y: a,F: a > int] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ( ( member_a @ Y @ S3 )
=> ( ord_less_eq_int @ ( F @ ( lattic6337796949162350289_a_int @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_782_arg__min__least,axiom,
! [S3: set_nat,Y: nat,F: nat > real] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S3 )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_783_arg__min__least,axiom,
! [S3: set_int,Y: int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ( ( member_int @ Y @ S3 )
=> ( ord_less_eq_real @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_784_arg__min__least,axiom,
! [S3: set_a,Y: a,F: a > real] :
( ( finite_finite_a @ S3 )
=> ( ( S3 != bot_bot_set_a )
=> ( ( member_a @ Y @ S3 )
=> ( ord_less_eq_real @ ( F @ ( lattic7288945864786915537a_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_785_arg__min__least,axiom,
! [S3: set_list_nat,Y: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( S3 != bot_bot_set_list_nat )
=> ( ( member_list_nat @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5785867957632790475at_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_786_finite__transitivity__chain,axiom,
! [A4: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
& ( R2 @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_787_finite__transitivity__chain,axiom,
! [A4: set_int,R2: int > int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X3: int] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: int,Y2: int,Z: int] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ A4 )
& ( R2 @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_788_finite__transitivity__chain,axiom,
! [A4: set_a,R2: a > a > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: a,Y2: a,Z: a] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [Y5: a] :
( ( member_a @ Y5 @ A4 )
& ( R2 @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_789_finite__transitivity__chain,axiom,
! [A4: set_list_nat,R2: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ! [X3: list_nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: list_nat,Y2: list_nat,Z: list_nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ? [Y5: list_nat] :
( ( member_list_nat @ Y5 @ A4 )
& ( R2 @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_list_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_790_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_791_subset__emptyI,axiom,
! [A4: set_int] :
( ! [X3: int] :
~ ( member_int @ X3 @ A4 )
=> ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_792_subset__emptyI,axiom,
! [A4: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_793_subset__emptyI,axiom,
! [A4: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A4 )
=> ( ord_le6045566169113846134st_nat @ A4 @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_794_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_795_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( member_nat @ N @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_796_unbounded__k__infinite,axiom,
! [K2: nat,S3: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K2 @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S3 ) ) )
=> ~ ( finite_finite_nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_797_finite__indexed__bound,axiom,
! [A4: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_798_finite__indexed__bound,axiom,
! [A4: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_799_finite__indexed__bound,axiom,
! [A4: set_int,P: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_800_finite__indexed__bound,axiom,
! [A4: set_a,P: a > int > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [X_12: int] : ( P @ X3 @ X_12 ) )
=> ? [M5: int] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ? [K: int] :
( ( ord_less_eq_int @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_801_finite__indexed__bound,axiom,
! [A4: set_nat,P: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [X_12: int] : ( P @ X3 @ X_12 ) )
=> ? [M5: int] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ? [K: int] :
( ( ord_less_eq_int @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_802_finite__indexed__bound,axiom,
! [A4: set_int,P: int > int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ? [X_12: int] : ( P @ X3 @ X_12 ) )
=> ? [M5: int] :
! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ? [K: int] :
( ( ord_less_eq_int @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_803_finite__indexed__bound,axiom,
! [A4: set_a,P: a > real > $o] :
( ( finite_finite_a @ A4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M5: real] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_804_finite__indexed__bound,axiom,
! [A4: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M5: real] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_805_finite__indexed__bound,axiom,
! [A4: set_int,P: int > real > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M5: real] :
! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_806_finite__indexed__bound,axiom,
! [A4: set_list_nat,P: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M5 )
& ( P @ X2 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_807_finite__enumerate__mono__iff,axiom,
! [S3: set_list_nat,M2: nat,N2: nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_nat @ M2 @ ( finite_card_list_nat @ S3 ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_list_nat @ S3 ) )
=> ( ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ M2 ) @ ( infini2033088105919815547st_nat @ S3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_808_finite__enumerate__mono__iff,axiom,
! [S3: set_nat,M2: nat,N2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S3 ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M2 ) @ ( infini8530281810654367211te_nat @ S3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_809_finite__enum__subset,axiom,
! [X6: set_list_nat,Y6: set_list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_list_nat @ X6 ) )
=> ( ( infini2033088105919815547st_nat @ X6 @ I2 )
= ( infini2033088105919815547st_nat @ Y6 @ I2 ) ) )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( finite8100373058378681591st_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ X6 ) @ ( finite_card_list_nat @ Y6 ) )
=> ( ord_le6045566169113846134st_nat @ X6 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_810_finite__enum__subset,axiom,
! [X6: set_nat,Y6: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
=> ( ( infini8530281810654367211te_nat @ X6 @ I2 )
= ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X6 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_811_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_812_Set_Ois__empty__def,axiom,
( is_empty_list_nat
= ( ^ [A6: set_list_nat] : ( A6 = bot_bot_set_list_nat ) ) ) ).
% Set.is_empty_def
thf(fact_813_finite__enumerate__mono,axiom,
! [M2: nat,N2: nat,S3: set_list_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_list_nat @ S3 ) )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ M2 ) @ ( infini2033088105919815547st_nat @ S3 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_814_finite__enumerate__mono,axiom,
! [M2: nat,N2: nat,S3: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M2 ) @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_815_enumerate__mono__iff,axiom,
! [S3: set_list_nat,M2: nat,N2: nat] :
( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ M2 ) @ ( infini2033088105919815547st_nat @ S3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_816_enumerate__mono__iff,axiom,
! [S3: set_nat,M2: nat,N2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M2 ) @ ( infini8530281810654367211te_nat @ S3 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_817_enumerate__mono__le__iff,axiom,
! [S3: set_list_nat,M2: nat,N2: nat] :
( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_eq_list_nat @ ( infini2033088105919815547st_nat @ S3 @ M2 ) @ ( infini2033088105919815547st_nat @ S3 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_818_enumerate__mono__le__iff,axiom,
! [S3: set_nat,M2: nat,N2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S3 @ M2 ) @ ( infini8530281810654367211te_nat @ S3 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_819_enumerate__in__set,axiom,
! [S3: set_list_nat,N2: nat] :
( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( member_list_nat @ ( infini2033088105919815547st_nat @ S3 @ N2 ) @ S3 ) ) ).
% enumerate_in_set
thf(fact_820_enumerate__in__set,axiom,
! [S3: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S3 @ N2 ) @ S3 ) ) ).
% enumerate_in_set
thf(fact_821_enumerate__Ex,axiom,
! [S3: set_nat,S: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ( member_nat @ S @ S3 )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S3 @ N3 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_822_le__enumerate,axiom,
! [S3: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ).
% le_enumerate
thf(fact_823_enumerate__mono,axiom,
! [M2: nat,N2: nat,S3: set_list_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ M2 ) @ ( infini2033088105919815547st_nat @ S3 @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_824_enumerate__mono,axiom,
! [M2: nat,N2: nat,S3: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ M2 ) @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_825_finite__enumerate__in__set,axiom,
! [S3: set_list_nat,N2: nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_list_nat @ S3 ) )
=> ( member_list_nat @ ( infini2033088105919815547st_nat @ S3 @ N2 ) @ S3 ) ) ) ).
% finite_enumerate_in_set
thf(fact_826_finite__enumerate__in__set,axiom,
! [S3: set_nat,N2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S3 @ N2 ) @ S3 ) ) ) ).
% finite_enumerate_in_set
thf(fact_827_finite__enumerate__Ex,axiom,
! [S3: set_list_nat,S: list_nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( member_list_nat @ S @ S3 )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_list_nat @ S3 ) )
& ( ( infini2033088105919815547st_nat @ S3 @ N3 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_828_finite__enumerate__Ex,axiom,
! [S3: set_nat,S: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( member_nat @ S @ S3 )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_nat @ S3 ) )
& ( ( infini8530281810654367211te_nat @ S3 @ N3 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_829_finite__enum__ext,axiom,
! [X6: set_list_nat,Y6: set_list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_list_nat @ X6 ) )
=> ( ( infini2033088105919815547st_nat @ X6 @ I2 )
= ( infini2033088105919815547st_nat @ Y6 @ I2 ) ) )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( finite8100373058378681591st_nat @ Y6 )
=> ( ( ( finite_card_list_nat @ X6 )
= ( finite_card_list_nat @ Y6 ) )
=> ( X6 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_830_finite__enum__ext,axiom,
! [X6: set_nat,Y6: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
=> ( ( infini8530281810654367211te_nat @ X6 @ I2 )
= ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X6 )
= ( finite_card_nat @ Y6 ) )
=> ( X6 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_831_finite__le__enumerate,axiom,
! [S3: set_nat,N2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_832_finite__enumerate__step,axiom,
! [S3: set_list_nat,N2: nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( ord_less_nat @ ( suc @ N2 ) @ ( finite_card_list_nat @ S3 ) )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ N2 ) @ ( infini2033088105919815547st_nat @ S3 @ ( suc @ N2 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_833_finite__enumerate__step,axiom,
! [S3: set_nat,N2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ ( suc @ N2 ) @ ( finite_card_nat @ S3 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N2 ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N2 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_834_card__gt__0__iff,axiom,
! [A4: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
= ( ( A4 != bot_bot_set_nat )
& ( finite_finite_nat @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_835_card__gt__0__iff,axiom,
! [A4: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
= ( ( A4 != bot_bot_set_int )
& ( finite_finite_int @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_836_card__gt__0__iff,axiom,
! [A4: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A4 ) )
= ( ( A4 != bot_bot_set_a )
& ( finite_finite_a @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_837_card__gt__0__iff,axiom,
! [A4: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
= ( ( A4 != bot_bot_set_list_nat )
& ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_838_Inf__fin_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B5 ) @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_839_Inf__fin_Osubset__imp,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( finite_finite_int @ B5 )
=> ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B5 ) @ ( lattic5235898064620869839in_int @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_840_Inf__fin_Osubset__imp,axiom,
! [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B5 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B5 ) @ ( lattic8209813465164889211_set_a @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_841_Inf__fin_Osubset__imp,axiom,
! [A4: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( finite_finite_real @ B5 )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ B5 ) @ ( lattic2677971596711400399n_real @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_842_Inf__fin_Osubset__imp,axiom,
! [A4: set_set_list_nat,B5: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A4 @ B5 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( finite7047420756378620717st_nat @ B5 )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ B5 ) @ ( lattic3683530169123051065st_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_843_Inf__fin_Osubset__imp,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ B5 )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ B5 ) @ ( lattic5191180550204456963st_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_844_Sup__fin_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_845_Sup__fin_Osubset__imp,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( finite_finite_int @ B5 )
=> ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A4 ) @ ( lattic1091506334969745077in_int @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_846_Sup__fin_Osubset__imp,axiom,
! [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B5 )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A4 ) @ ( lattic2918178356826803221_set_a @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_847_Sup__fin_Osubset__imp,axiom,
! [A4: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( finite_finite_real @ B5 )
=> ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A4 ) @ ( lattic8928443293348198069n_real @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_848_Sup__fin_Osubset__imp,axiom,
! [A4: set_set_list_nat,B5: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A4 @ B5 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( finite7047420756378620717st_nat @ B5 )
=> ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A4 ) @ ( lattic2169124122975652127st_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_849_Sup__fin_Osubset__imp,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ B5 )
=> ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A4 ) @ ( lattic6411832703407573737st_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_850_card__0__eq,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_851_card__0__eq,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( ( finite_card_int @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_852_card__0__eq,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( ( finite_card_a @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_853_card__0__eq,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( ( finite_card_list_nat @ A4 )
= zero_zero_nat )
= ( A4 = bot_bot_set_list_nat ) ) ) ).
% card_0_eq
thf(fact_854_Max__mono,axiom,
! [M4: set_int,N5: set_int] :
( ( ord_less_eq_set_int @ M4 @ N5 )
=> ( ( M4 != bot_bot_set_int )
=> ( ( finite_finite_int @ N5 )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M4 ) @ ( lattic8263393255366662781ax_int @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_855_Max__mono,axiom,
! [M4: set_real,N5: set_real] :
( ( ord_less_eq_set_real @ M4 @ N5 )
=> ( ( M4 != bot_bot_set_real )
=> ( ( finite_finite_real @ N5 )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ M4 ) @ ( lattic4275903605611617917x_real @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_856_Max__mono,axiom,
! [M4: set_list_nat,N5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ M4 @ N5 )
=> ( ( M4 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ N5 )
=> ( ord_less_eq_list_nat @ ( lattic2817244848751514289st_nat @ M4 ) @ ( lattic2817244848751514289st_nat @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_857_Max__mono,axiom,
! [M4: set_nat,N5: set_nat] :
( ( ord_less_eq_set_nat @ M4 @ N5 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N5 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M4 ) @ ( lattic8265883725875713057ax_nat @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_858_Max_Osubset__imp,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( finite_finite_int @ B5 )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ ( lattic8263393255366662781ax_int @ B5 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_859_Max_Osubset__imp,axiom,
! [A4: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( finite_finite_real @ B5 )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ ( lattic4275903605611617917x_real @ B5 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_860_Max_Osubset__imp,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ B5 )
=> ( ord_less_eq_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ ( lattic2817244848751514289st_nat @ B5 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_861_Max_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ B5 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_862_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_863_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_864_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_865_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_866_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_867_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_868_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_869_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_870_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_871_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_872_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_873_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_874_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_875_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_876_card_Oinfinite,axiom,
! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( ( finite_card_a @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_877_card_Oinfinite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_card_nat @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_878_card_Oinfinite,axiom,
! [A4: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A4 )
=> ( ( finite_card_list_nat @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_879_card_Oinfinite,axiom,
! [A4: set_int] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_card_int @ A4 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_880_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_881_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_882_Max_Obounded__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ X )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_eq_list_nat @ X4 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_883_Max_Obounded__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ X4 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_884_Max_Obounded__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ X )
= ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ X4 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_885_Max_Obounded__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_886_Max__less__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ X )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_list_nat @ X4 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_887_Max__less__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_int @ X4 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_888_Max__less__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic4275903605611617917x_real @ A4 ) @ X )
= ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_real @ X4 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_889_Max__less__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_890_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_891_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_892_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_893_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_894_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_895_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_896_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_897_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_898_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_899_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_900_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_901_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_902_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_903_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_904_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_905_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_906_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_907_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_908_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_909_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_910_Sup__fin__Max,axiom,
lattic1093996805478795353in_nat = lattic8265883725875713057ax_nat ).
% Sup_fin_Max
thf(fact_911_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_912_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_913_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_914_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_915_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_916_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_917_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_918_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_919_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_920_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_921_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_922_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_923_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_924_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_925_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_926_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_927_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_928_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_929_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_930_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_931_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_932_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_933_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_934_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_935_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_936_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_937_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I2 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_938_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_939_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_940_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_941_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_942_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_943_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_944_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_945_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_946_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_947_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_948_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_949_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_950_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_951_card__le__Suc0__iff__eq,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A4 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ A4 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_952_card__le__Suc0__iff__eq,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A4 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_953_card__le__Suc0__iff__eq,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A4 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_954_card__le__Suc0__iff__eq,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ! [Y4: int] :
( ( member_int @ Y4 @ A4 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_955_Inf__fin__le__Sup__fin,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ A4 ) @ ( lattic6411832703407573737st_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_956_Inf__fin__le__Sup__fin,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_957_Inf__fin__le__Sup__fin,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A4 ) @ ( lattic1091506334969745077in_int @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_958_Inf__fin__le__Sup__fin,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A4 ) @ ( lattic2918178356826803221_set_a @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_959_Inf__fin__le__Sup__fin,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ A4 ) @ ( lattic8928443293348198069n_real @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_960_Inf__fin__le__Sup__fin,axiom,
! [A4: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ A4 ) @ ( lattic2169124122975652127st_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_961_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_962_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_963_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_964_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_965_lift__Suc__mono__le,axiom,
! [F: nat > set_list_nat,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_966_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_967_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_968_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_set_a @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_969_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_real @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_970_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_nat,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_971_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_972_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_973_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_974_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_975_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_976_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_977_Max_OcoboundedI,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ord_less_eq_list_nat @ A @ ( lattic2817244848751514289st_nat @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_978_Max_OcoboundedI,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ord_less_eq_int @ A @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_979_Max_OcoboundedI,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ord_less_eq_real @ A @ ( lattic4275903605611617917x_real @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_980_Max_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_981_Max__eq__if,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( finite8100373058378681591st_nat @ B5 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B5 )
& ( ord_less_eq_list_nat @ X3 @ Xa ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B5 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ A4 )
& ( ord_less_eq_list_nat @ X3 @ Xa ) ) )
=> ( ( lattic2817244848751514289st_nat @ A4 )
= ( lattic2817244848751514289st_nat @ B5 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_982_Max__eq__if,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ? [Xa: int] :
( ( member_int @ Xa @ A4 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ( lattic8263393255366662781ax_int @ A4 )
= ( lattic8263393255366662781ax_int @ B5 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_983_Max__eq__if,axiom,
! [A4: set_real,B5: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_real @ B5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B5 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ? [Xa: real] :
( ( member_real @ Xa @ A4 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ( lattic4275903605611617917x_real @ A4 )
= ( lattic4275903605611617917x_real @ B5 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_984_Max__eq__if,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A4 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= ( lattic8265883725875713057ax_nat @ B5 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_985_Max__eqI,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ A4 )
=> ( ord_less_eq_list_nat @ Y2 @ X ) )
=> ( ( member_list_nat @ X @ A4 )
=> ( ( lattic2817244848751514289st_nat @ A4 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_986_Max__eqI,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A4 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( member_int @ X @ A4 )
=> ( ( lattic8263393255366662781ax_int @ A4 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_987_Max__eqI,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ A4 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( ( lattic4275903605611617917x_real @ A4 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_988_Max__eqI,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A4 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( member_nat @ X @ A4 )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_989_Max__ge,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ X @ A4 )
=> ( ord_less_eq_list_nat @ X @ ( lattic2817244848751514289st_nat @ A4 ) ) ) ) ).
% Max_ge
thf(fact_990_Max__ge,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) ) ) ) ).
% Max_ge
thf(fact_991_Max__ge,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A4 ) ) ) ) ).
% Max_ge
thf(fact_992_Max__ge,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).
% Max_ge
thf(fact_993_Max__in,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( member_int @ ( lattic8263393255366662781ax_int @ A4 ) @ A4 ) ) ) ).
% Max_in
thf(fact_994_Max__in,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( member_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ A4 ) ) ) ).
% Max_in
thf(fact_995_Max__in,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ).
% Max_in
thf(fact_996_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_997_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_998_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_999_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1000_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1001_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1002_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1003_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1004_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1005_Sup__fin_OcoboundedI,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ord_less_eq_list_nat @ A @ ( lattic6411832703407573737st_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1006_Sup__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1007_Sup__fin_OcoboundedI,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ord_less_eq_int @ A @ ( lattic1091506334969745077in_int @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1008_Sup__fin_OcoboundedI,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A @ A4 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1009_Sup__fin_OcoboundedI,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ord_less_eq_real @ A @ ( lattic8928443293348198069n_real @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1010_Sup__fin_OcoboundedI,axiom,
! [A4: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( member_set_list_nat @ A @ A4 )
=> ( ord_le6045566169113846134st_nat @ A @ ( lattic2169124122975652127st_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1011_Inf__fin_OcoboundedI,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1012_Inf__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1013_Inf__fin_OcoboundedI,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1014_Inf__fin_OcoboundedI,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A @ A4 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1015_Inf__fin_OcoboundedI,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1016_Inf__fin_OcoboundedI,axiom,
! [A4: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( member_set_list_nat @ A @ A4 )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1017_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1018_Max_OboundedI,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ord_less_eq_list_nat @ A3 @ X ) )
=> ( ord_less_eq_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1019_Max_OboundedI,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_int @ A3 @ X ) )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1020_Max_OboundedI,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_real @ A3 @ X ) )
=> ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1021_Max_OboundedI,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ A3 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1022_Max_OboundedE,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic2817244848751514289st_nat @ A4 ) @ X )
=> ! [A8: list_nat] :
( ( member_list_nat @ A8 @ A4 )
=> ( ord_less_eq_list_nat @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1023_Max_OboundedE,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A4 ) @ X )
=> ! [A8: int] :
( ( member_int @ A8 @ A4 )
=> ( ord_less_eq_int @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1024_Max_OboundedE,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A4 ) @ X )
=> ! [A8: real] :
( ( member_real @ A8 @ A4 )
=> ( ord_less_eq_real @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1025_Max_OboundedE,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1026_eq__Max__iff,axiom,
! [A4: set_list_nat,M2: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( M2
= ( lattic2817244848751514289st_nat @ A4 ) )
= ( ( member_list_nat @ M2 @ A4 )
& ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_eq_list_nat @ X4 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1027_eq__Max__iff,axiom,
! [A4: set_int,M2: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( M2
= ( lattic8263393255366662781ax_int @ A4 ) )
= ( ( member_int @ M2 @ A4 )
& ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ X4 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1028_eq__Max__iff,axiom,
! [A4: set_real,M2: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( M2
= ( lattic4275903605611617917x_real @ A4 ) )
= ( ( member_real @ M2 @ A4 )
& ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ X4 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1029_eq__Max__iff,axiom,
! [A4: set_nat,M2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( M2
= ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ( member_nat @ M2 @ A4 )
& ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1030_Max__ge__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ X @ ( lattic2817244848751514289st_nat @ A4 ) )
= ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
& ( ord_less_eq_list_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1031_Max__ge__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ X @ X4 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1032_Max__ge__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A4 ) )
= ( ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ X @ X4 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1033_Max__ge__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1034_Max__eq__iff,axiom,
! [A4: set_list_nat,M2: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ( lattic2817244848751514289st_nat @ A4 )
= M2 )
= ( ( member_list_nat @ M2 @ A4 )
& ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_eq_list_nat @ X4 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1035_Max__eq__iff,axiom,
! [A4: set_int,M2: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ( lattic8263393255366662781ax_int @ A4 )
= M2 )
= ( ( member_int @ M2 @ A4 )
& ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ X4 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1036_Max__eq__iff,axiom,
! [A4: set_real,M2: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ( lattic4275903605611617917x_real @ A4 )
= M2 )
= ( ( member_real @ M2 @ A4 )
& ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ X4 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1037_Max__eq__iff,axiom,
! [A4: set_nat,M2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A4 )
= M2 )
= ( ( member_nat @ M2 @ A4 )
& ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1038_Max__gr__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_list_nat @ X @ ( lattic2817244848751514289st_nat @ A4 ) )
= ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
& ( ord_less_list_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_1039_Max__gr__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A4 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_int @ X @ X4 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_1040_Max__gr__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic4275903605611617917x_real @ A4 ) )
= ( ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_real @ X @ X4 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_1041_Max__gr__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_1042_enumerate__step,axiom,
! [S3: set_list_nat,N2: nat] :
( ~ ( finite8100373058378681591st_nat @ S3 )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S3 @ N2 ) @ ( infini2033088105919815547st_nat @ S3 @ ( suc @ N2 ) ) ) ) ).
% enumerate_step
thf(fact_1043_enumerate__step,axiom,
! [S3: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N2 ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N2 ) ) ) ) ).
% enumerate_step
thf(fact_1044_card__eq__0__iff,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_1045_card__eq__0__iff,axiom,
! [A4: set_int] :
( ( ( finite_card_int @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_int )
| ~ ( finite_finite_int @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_1046_card__eq__0__iff,axiom,
! [A4: set_a] :
( ( ( finite_card_a @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_a )
| ~ ( finite_finite_a @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_1047_card__eq__0__iff,axiom,
! [A4: set_list_nat] :
( ( ( finite_card_list_nat @ A4 )
= zero_zero_nat )
= ( ( A4 = bot_bot_set_list_nat )
| ~ ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_1048_Sup__fin_Obounded__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A4 ) @ X )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_eq_list_nat @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1049_Sup__fin_Obounded__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1050_Sup__fin_Obounded__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A4 ) @ X )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1051_Sup__fin_Obounded__iff,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A4 ) @ X )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( ord_less_eq_set_a @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1052_Sup__fin_Obounded__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A4 ) @ X )
= ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1053_Sup__fin_Obounded__iff,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A4 ) @ X )
= ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A4 )
=> ( ord_le6045566169113846134st_nat @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1054_Inf__fin_Obounded__iff,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A4 ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( ord_less_eq_list_nat @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1055_Inf__fin_Obounded__iff,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1056_Inf__fin_Obounded__iff,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A4 ) )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1057_Inf__fin_Obounded__iff,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A4 ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( ord_less_eq_set_a @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1058_Inf__fin_Obounded__iff,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A4 ) )
= ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1059_Inf__fin_Obounded__iff,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A4 ) )
= ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A4 )
=> ( ord_le6045566169113846134st_nat @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1060_Sup__fin_OboundedI,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ord_less_eq_list_nat @ A3 @ X ) )
=> ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1061_Sup__fin_OboundedI,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ A3 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1062_Sup__fin_OboundedI,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_int @ A3 @ X ) )
=> ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1063_Sup__fin_OboundedI,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A4 )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1064_Sup__fin_OboundedI,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_real @ A3 @ X ) )
=> ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1065_Sup__fin_OboundedI,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ! [A3: set_list_nat] :
( ( member_set_list_nat @ A3 @ A4 )
=> ( ord_le6045566169113846134st_nat @ A3 @ X ) )
=> ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1066_Sup__fin_OboundedE,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A4 ) @ X )
=> ! [A8: list_nat] :
( ( member_list_nat @ A8 @ A4 )
=> ( ord_less_eq_list_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1067_Sup__fin_OboundedE,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1068_Sup__fin_OboundedE,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A4 ) @ X )
=> ! [A8: int] :
( ( member_int @ A8 @ A4 )
=> ( ord_less_eq_int @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1069_Sup__fin_OboundedE,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A4 ) @ X )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A4 )
=> ( ord_less_eq_set_a @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1070_Sup__fin_OboundedE,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A4 ) @ X )
=> ! [A8: real] :
( ( member_real @ A8 @ A4 )
=> ( ord_less_eq_real @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1071_Sup__fin_OboundedE,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A4 ) @ X )
=> ! [A8: set_list_nat] :
( ( member_set_list_nat @ A8 @ A4 )
=> ( ord_le6045566169113846134st_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1072_Inf__fin_OboundedI,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ord_less_eq_list_nat @ X @ A3 ) )
=> ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1073_Inf__fin_OboundedI,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ X @ A3 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1074_Inf__fin_OboundedI,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_int @ X @ A3 ) )
=> ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1075_Inf__fin_OboundedI,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A4 )
=> ( ord_less_eq_set_a @ X @ A3 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1076_Inf__fin_OboundedI,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_real @ X @ A3 ) )
=> ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1077_Inf__fin_OboundedI,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ! [A3: set_list_nat] :
( ( member_set_list_nat @ A3 @ A4 )
=> ( ord_le6045566169113846134st_nat @ X @ A3 ) )
=> ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1078_Inf__fin_OboundedE,axiom,
! [A4: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( A4 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A4 ) )
=> ! [A8: list_nat] :
( ( member_list_nat @ A8 @ A4 )
=> ( ord_less_eq_list_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1079_Inf__fin_OboundedE,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A4 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1080_Inf__fin_OboundedE,axiom,
! [A4: set_int,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A4 ) )
=> ! [A8: int] :
( ( member_int @ A8 @ A4 )
=> ( ord_less_eq_int @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1081_Inf__fin_OboundedE,axiom,
! [A4: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A4 ) )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A4 )
=> ( ord_less_eq_set_a @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1082_Inf__fin_OboundedE,axiom,
! [A4: set_real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A4 ) )
=> ! [A8: real] :
( ( member_real @ A8 @ A4 )
=> ( ord_less_eq_real @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1083_Inf__fin_OboundedE,axiom,
! [A4: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A4 )
=> ( ( A4 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A4 ) )
=> ! [A8: set_list_nat] :
( ( member_set_list_nat @ A8 @ A4 )
=> ( ord_le6045566169113846134st_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1084_card__ge__0__finite,axiom,
! [A4: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A4 ) )
=> ( finite_finite_a @ A4 ) ) ).
% card_ge_0_finite
thf(fact_1085_card__ge__0__finite,axiom,
! [A4: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
=> ( finite_finite_nat @ A4 ) ) ).
% card_ge_0_finite
thf(fact_1086_card__ge__0__finite,axiom,
! [A4: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
=> ( finite8100373058378681591st_nat @ A4 ) ) ).
% card_ge_0_finite
thf(fact_1087_card__ge__0__finite,axiom,
! [A4: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
=> ( finite_finite_int @ A4 ) ) ).
% card_ge_0_finite
thf(fact_1088_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1089_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1090_list__incr__def,axiom,
( list_incr
= ( ^ [I4: nat,X4: list_nat] : ( list_update_nat @ X4 @ I4 @ ( suc @ ( nth_nat @ X4 @ I4 ) ) ) ) ) ).
% list_incr_def
thf(fact_1091_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1092_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1093_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1094_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1095_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1096_aA__idx__eq,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ( nth_a @ ( aA_a @ a2 ) @ ( counta3566351752493190365t_on_a @ a2 @ A ) )
= A ) ) ).
% aA_idx_eq
thf(fact_1097_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1098_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1099_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1100_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1101_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1102_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1103_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_1104_set__aA,axiom,
( ( set_a2 @ ( aA_a @ a2 ) )
= a2 ) ).
% set_aA
thf(fact_1105_nth__aA__in__G,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ a2 ) )
=> ( member_a @ ( nth_a @ ( aA_a @ a2 ) @ I ) @ g ) ) ).
% nth_aA_in_G
thf(fact_1106_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1107_AsubG,axiom,
ord_less_eq_set_a @ a2 @ g ).
% AsubG
thf(fact_1108_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1109_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1110_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1111_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1112_length__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I @ X ) )
= ( size_size_list_nat @ X ) ) ).
% length_list_incr
thf(fact_1113_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1114_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1115_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1116_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1117_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1118_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1119_le__nat__iff,axiom,
! [K2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_1120_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1121_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1122_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1123_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X4: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X4 )
= ( size_size_list_nat @ Y4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ X4 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X4 @ I4 ) @ ( nth_nat @ Y4 @ I4 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_1124_pointwise__le__refl,axiom,
! [X: list_nat] : ( pointwise_le @ X @ X ) ).
% pointwise_le_refl
thf(fact_1125_list__incr__Cons,axiom,
! [I: nat,K2: nat,Ks: list_nat] :
( ( list_incr @ ( suc @ I ) @ ( cons_nat @ K2 @ Ks ) )
= ( cons_nat @ K2 @ ( list_incr @ I @ Ks ) ) ) ).
% list_incr_Cons
thf(fact_1126_sum__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X ) )
=> ( ( groups4561878855575611511st_nat @ ( list_incr @ I @ X ) )
= ( suc @ ( groups4561878855575611511st_nat @ X ) ) ) ) ).
% sum_list_incr
thf(fact_1127_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_1128_pointwise__le__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( pointwise_le @ X @ Y )
=> ( ( pointwise_le @ Y @ X )
=> ( X = Y ) ) ) ).
% pointwise_le_antisym
thf(fact_1129_pointwise__le__trans,axiom,
! [X: list_nat,Y: list_nat,Z3: list_nat] :
( ( pointwise_le @ X @ Y )
=> ( ( pointwise_le @ Y @ Z3 )
=> ( pointwise_le @ X @ Z3 ) ) ) ).
% pointwise_le_trans
thf(fact_1130_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1131_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1132_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1133_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1134_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1135_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X4: list_nat,Y4: list_nat] :
( ( pointwise_le @ X4 @ Y4 )
& ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_size_list_nat @ X4 ) )
& ( ord_less_nat @ ( nth_nat @ X4 @ K3 ) @ ( nth_nat @ Y4 @ K3 ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_1136_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X4: list_nat,Y4: list_nat] :
( ( pointwise_le @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% pointwise_less_def
thf(fact_1137_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X4: list_nat,Y4: list_nat] :
( ( pointwise_less @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_1138_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_1139_min__pointwise__ge__iff,axiom,
! [U2: set_list_nat,R: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U3: list_nat] :
( ( member_list_nat @ U3 @ U2 )
=> ( ( size_size_list_nat @ U3 )
= R ) )
=> ( ( ( size_size_list_nat @ X )
= R )
=> ( ( pointwise_le @ X @ ( min_pointwise @ R @ U2 ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ U2 )
=> ( pointwise_le @ X @ X4 ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_1140_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_1141_max__pointwise__le__iff,axiom,
! [U2: set_list_nat,R: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U3: list_nat] :
( ( member_list_nat @ U3 @ U2 )
=> ( ( size_size_list_nat @ U3 )
= R ) )
=> ( ( ( size_size_list_nat @ X )
= R )
=> ( ( pointwise_le @ ( max_pointwise @ R @ U2 ) @ X )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ U2 )
=> ( pointwise_le @ X4 @ X ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_1142_max__pointwise__mono,axiom,
! [X8: set_list_nat,X6: set_list_nat,R: nat] :
( ( ord_le6045566169113846134st_nat @ X8 @ X6 )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( X8 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R @ X8 ) @ ( max_pointwise @ R @ X6 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_1143_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_1144_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_1145_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1146_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1147_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1148_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1149_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1150_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_1151_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_1152_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_1153_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1154_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1155_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1156_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1157_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_1158_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1159_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1160_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_1161_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1162_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1163_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1164_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_1165_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1166_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1167_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1168_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_1169_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1170_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_1171_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1172_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1173_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1174_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1175_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1176_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1177_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1178_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1179_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1180_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_1181_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1182_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1183_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1184_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1185_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1186_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_1187_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1188_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1189_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1190_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1191_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1192_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1193_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1194_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1195_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1196_pairwise__minus__cancel,axiom,
! [Z3: list_nat,X: list_nat,Y: list_nat] :
( ( pointwise_le @ Z3 @ X )
=> ( ( pointwise_le @ Z3 @ Y )
=> ( ( ( minus_minus_list_nat @ X @ Z3 )
= ( minus_minus_list_nat @ Y @ Z3 ) )
=> ( X = Y ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_1197_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1198_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1199_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1200_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1201_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1202_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1203_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1204_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1205_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1206_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1207_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1208_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1209_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1210_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1211_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_1212_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1213_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1214_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1215_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1216_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1217_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1218_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1219_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1220_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1221_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1222_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1223_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1224_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1225_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1226_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1227_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1228_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1229_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1230_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1231_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: 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 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1232_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_1233_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1234_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1235_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1236_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1237_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1238_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1239_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_1240_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1241_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1242_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1243_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1244_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1245_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1246_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1247_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1248_int__minus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1249_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_1250_sum__list__minus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_1251_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1252_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_1253_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1254_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1255_Suc__as__int,axiom,
( suc
= ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1256_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1257_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1258_infinite__int__iff__unbounded,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M: int] :
? [N: int] :
( ( ord_less_int @ M @ ( abs_abs_int @ N ) )
& ( member_int @ N @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_1259_infinite__int__iff__unbounded__le,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M: int] :
? [N: int] :
( ( ord_less_eq_int @ M @ ( abs_abs_int @ N ) )
& ( member_int @ N @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_1260_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1261_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1262_nat__intermed__int__val,axiom,
! [M2: nat,N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1263_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1264_incr__lemma,axiom,
! [D2: int,Z3: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z3 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1265_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1266_plusinfinity,axiom,
! [D2: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
ord_less_nat @ i @ ( finite_card_a @ a2 ) ).
thf(conj_1,conjecture,
ord_less_eq_nat @ ( nth_nat @ x @ i ) @ ( nth_nat @ ( list_incr @ ( counta3566351752493190365t_on_a @ a2 @ a3 ) @ x ) @ i ) ).
%------------------------------------------------------------------------------