TPTP Problem File: SLH0054^1.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 : Frequency_Moments/0087_Frequency_Moment_k/prob_00623_026891__20029820_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1190 ( 426 unt; 141 typ; 0 def)
% Number of atoms : 3181 (1025 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 8950 ( 494 ~; 58 |; 270 &;6474 @)
% ( 0 <=>;1654 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 465 ( 465 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 126 usr; 21 con; 0-2 aty)
% Number of variables : 2737 ( 115 ^;2449 !; 173 ?;2737 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:20:04.271
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_Pr6218003697084177305l_real: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Real__Oreal_J,type,
multiset_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (126)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
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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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Frequency__Moment__k_OM_092_060_094sub_0621,type,
frequency_Moment_M_1: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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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_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > 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,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_List_Olinorder_Osorted__list__of__set_001t__Real__Oreal,type,
sorted6366500744023230182t_real: ( real > real > $o ) > set_real > list_real ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
linord4252657396651189596t_real: set_real > list_real ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olist_Ohd_001t__Real__Oreal,type,
hd_real: list_real > real ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_e8644085759156585930at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Omember_001t__Real__Oreal,type,
member_real: list_real > real > $o ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onths_001t__Real__Oreal,type,
nths_real: list_real > set_nat > list_real ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Real__Oreal,type,
remdups_adj_real: list_real > list_real ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
sorted5214655850825725294at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Real__Oreal,type,
linord36121425647212990t_real: multiset_real > list_real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__List__Olist_It__Nat__Onat_J,type,
bot_bot_list_nat: list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
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thf(sy_c_Relation_ODomain_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
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thf(sy_c_Relation_OId__on_001t__Real__Oreal,type,
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thf(sy_c_Relation_ORange_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
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thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
is_emp1662574758705540307at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
is_empty_real: set_real > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
is_singleton_real: set_real > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
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_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_v_as,type,
as: list_nat ).
thf(sy_v_k,type,
k: nat ).
% Relevant facts (1048)
thf(fact_0_False,axiom,
as != nil_nat ).
% False
thf(fact_1_mem__simps_I2_J,axiom,
! [C: real] :
~ ( member_real2 @ C @ bot_bot_set_real ) ).
% mem_simps(2)
thf(fact_2_mem__simps_I2_J,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% mem_simps(2)
thf(fact_3_mem__simps_I2_J,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% mem_simps(2)
thf(fact_4_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X: real] :
~ ( member_real2 @ X @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_5_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat2 @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_6_all__not__in__conv,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ! [X: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X @ A ) )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_7_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X: real] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_8_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_9_Collect__empty__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( ! [X: product_prod_nat_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_10_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X: real] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_11_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_12_empty__Collect__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ P ) )
= ( ! [X: product_prod_nat_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_13_emptyE,axiom,
! [A2: real] :
~ ( member_real2 @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_14_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_15_emptyE,axiom,
! [A2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_16_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real2 @ A2 @ A ) ) ).
% equals0D
thf(fact_17_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_18_equals0D,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( A = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_19_equals0I,axiom,
! [A: set_real] :
( ! [Y: real] :
~ ( member_real2 @ Y @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_20_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat2 @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_21_equals0I,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [Y: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y @ A )
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_22_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X: real] : ( member_real2 @ X @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_23_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat2 @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_24_ex__in__conv,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A ) )
= ( A != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_25_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_26_set__empty,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( set_Pr5648618587558075414at_nat @ Xs )
= bot_bo2099793752762293965at_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% set_empty
thf(fact_27_set__empty,axiom,
! [Xs: list_real] :
( ( ( set_real2 @ Xs )
= bot_bot_set_real )
= ( Xs = nil_real ) ) ).
% set_empty
thf(fact_28_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_29_set__empty2,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% set_empty2
thf(fact_30_set__empty2,axiom,
! [Xs: list_real] :
( ( bot_bot_set_real
= ( set_real2 @ Xs ) )
= ( Xs = nil_real ) ) ).
% set_empty2
thf(fact_31_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_32_bot__set__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% bot_set_def
thf(fact_33_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_34_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_35_empty__set,axiom,
( bot_bo2099793752762293965at_nat
= ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).
% empty_set
thf(fact_36_empty__set,axiom,
( bot_bot_set_real
= ( set_real2 @ nil_real ) ) ).
% empty_set
thf(fact_37_non__empty__space,axiom,
( ( as != nil_nat )
=> ( ( frequency_Moment_M_1 @ as )
!= bot_bo2099793752762293965at_nat ) ) ).
% non_empty_space
thf(fact_38_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_39_Set_Ois__empty__def,axiom,
( is_emp1662574758705540307at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat] : ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).
% Set.is_empty_def
thf(fact_40_Set_Ois__empty__def,axiom,
( is_empty_real
= ( ^ [A3: set_real] : ( A3 = bot_bot_set_real ) ) ) ).
% Set.is_empty_def
thf(fact_41_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_42_nths__empty,axiom,
! [Xs: list_nat] :
( ( nths_nat @ Xs @ bot_bot_set_nat )
= nil_nat ) ).
% nths_empty
thf(fact_43_bot__list__def,axiom,
bot_bot_list_nat = nil_nat ).
% bot_list_def
thf(fact_44_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_45_Collect__empty__eq__bot,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( P = bot_bo482883023278783056_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_46_Collect__empty__eq__bot,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( P = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_47_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_48_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_49_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X: real] : ( member_real2 @ X @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_50_in__set__member,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
= ( member_real @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_51_in__set__member,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( member6104210405413575452at_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_52_in__set__member,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_53_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_54_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_55_nths__nil,axiom,
! [A: set_nat] :
( ( nths_nat @ nil_nat @ A )
= nil_nat ) ).
% nths_nil
thf(fact_56_in__set__nthsD,axiom,
! [X2: real,Xs: list_real,I: set_nat] :
( ( member_real2 @ X2 @ ( set_real2 @ ( nths_real @ Xs @ I ) ) )
=> ( member_real2 @ X2 @ ( set_real2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_57_in__set__nthsD,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,I: set_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( nths_P6079298444859966469at_nat @ Xs @ I ) ) )
=> ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_58_in__set__nthsD,axiom,
! [X2: nat,Xs: list_nat,I: set_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_59_notin__set__nthsI,axiom,
! [X2: real,Xs: list_real,I: set_nat] :
( ~ ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ~ ( member_real2 @ X2 @ ( set_real2 @ ( nths_real @ Xs @ I ) ) ) ) ).
% notin_set_nthsI
thf(fact_60_notin__set__nthsI,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,I: set_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( nths_P6079298444859966469at_nat @ Xs @ I ) ) ) ) ).
% notin_set_nthsI
thf(fact_61_notin__set__nthsI,axiom,
! [X2: nat,Xs: list_nat,I: set_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) ) ) ).
% notin_set_nthsI
thf(fact_62_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P2: real > $o,Xs2: list_real] :
? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs2 ) )
& ( P2 @ X )
& ! [Y3: real] :
( ( ( member_real2 @ Y3 @ ( set_real2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_63_list__ex1__iff,axiom,
( list_e8644085759156585930at_nat
= ( ^ [P2: product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat] :
? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
& ( P2 @ X )
& ! [Y3: product_prod_nat_nat] :
( ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_64_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_65_fin__space,axiom,
( ( as != nil_nat )
=> ( finite6177210948735845034at_nat @ ( frequency_Moment_M_1 @ as ) ) ) ).
% fin_space
thf(fact_66_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_67_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_68_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Id_on_empty
thf(fact_69_Id__on__empty,axiom,
( ( id_on_2554058798563519774at_nat @ bot_bo2099793752762293965at_nat )
= bot_bo5327735625951526323at_nat ) ).
% Id_on_empty
thf(fact_70_Id__on__empty,axiom,
( ( id_on_real @ bot_bot_set_real )
= bot_bo3948376660626123781l_real ) ).
% Id_on_empty
thf(fact_71_Range__empty,axiom,
( ( range_nat_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Range_empty
thf(fact_72_Domain__empty,axiom,
( ( domain_nat_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Domain_empty
thf(fact_73_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_74_card__space,axiom,
( ( as != nil_nat )
=> ( ( finite711546835091564841at_nat @ ( frequency_Moment_M_1 @ as ) )
= ( size_size_list_nat @ as ) ) ) ).
% card_space
thf(fact_75_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real2 @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_81_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_82_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_83_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_84_in__set__insert,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_85_in__set__insert,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( insert8944034826898310173at_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_86_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_87_List_Ofinite__set,axiom,
! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_88_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_89_finite__Domain,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R )
=> ( finite_finite_nat @ ( domain_nat_nat @ R ) ) ) ).
% finite_Domain
thf(fact_90_finite__Range,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R )
=> ( finite_finite_nat @ ( range_nat_nat @ R ) ) ) ).
% finite_Range
thf(fact_91_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_92_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_93_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X: nat] :
( ( member_nat2 @ X @ A3 )
& ( P2 @ X )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_94_can__select__def,axiom,
( can_select_real
= ( ^ [P2: real > $o,A3: set_real] :
? [X: real] :
( ( member_real2 @ X @ A3 )
& ( P2 @ X )
& ! [Y3: real] :
( ( ( member_real2 @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_95_can__select__def,axiom,
( can_se4754832747099445502at_nat
= ( ^ [P2: product_prod_nat_nat > $o,A3: set_Pr1261947904930325089at_nat] :
? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A3 )
& ( P2 @ X )
& ! [Y3: product_prod_nat_nat] :
( ( ( member8440522571783428010at_nat @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_96_finite__list,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ? [Xs3: list_P6011104703257516679at_nat] :
( ( set_Pr5648618587558075414at_nat @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_97_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_98_Domain__empty__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( ( domain_nat_nat @ R )
= bot_bot_set_nat )
= ( R = bot_bo2099793752762293965at_nat ) ) ).
% Domain_empty_iff
thf(fact_99_Range__empty__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( ( range_nat_nat @ R )
= bot_bot_set_nat )
= ( R = bot_bo2099793752762293965at_nat ) ) ).
% Range_empty_iff
thf(fact_100_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_101_null__def,axiom,
( null_nat
= ( ^ [Xs2: list_nat] : ( Xs2 = nil_nat ) ) ) ).
% null_def
thf(fact_102_finite__transitivity__chain,axiom,
! [A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z: nat] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ? [Y4: nat] :
( ( member_nat2 @ Y4 @ A )
& ( R2 @ X3 @ Y4 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_103_finite__transitivity__chain,axiom,
! [A: set_Pr1261947904930325089at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [X3: product_prod_nat_nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
=> ? [Y4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y4 @ A )
& ( R2 @ X3 @ Y4 ) ) )
=> ( A = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_104_finite__transitivity__chain,axiom,
! [A: set_real,R2: real > real > $o] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: real,Y: real,Z: real] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ A )
=> ? [Y4: real] :
( ( member_real2 @ Y4 @ A )
& ( R2 @ X3 @ Y4 ) ) )
=> ( A = bot_bot_set_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_105_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_106_infinite__imp__nonempty,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S )
=> ( S != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_107_infinite__imp__nonempty,axiom,
! [S: set_real] :
( ~ ( finite_finite_real @ S )
=> ( S != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_108_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_109_finite_OemptyI,axiom,
finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% finite.emptyI
thf(fact_110_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_111_sorted__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set_eq_Nil_iff
thf(fact_112_sorted__list__of__set__eq__Nil__iff,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( ( linord4252657396651189596t_real @ A )
= nil_real )
= ( A = bot_bot_set_real ) ) ) ).
% sorted_list_of_set_eq_Nil_iff
thf(fact_113_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_114_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_115_card__length,axiom,
! [Xs: list_P6011104703257516679at_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) @ ( size_s5460976970255530739at_nat @ Xs ) ) ).
% card_length
thf(fact_116_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_117_order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% order.refl
thf(fact_118_order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% order.refl
thf(fact_119_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_120_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_121_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_122_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord4252657396651189596t_real @ bot_bot_set_real )
= nil_real ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_123_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_124_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_125_length__sorted__list__of__set,axiom,
! [A: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A ) )
= ( finite_card_nat @ A ) ) ).
% length_sorted_list_of_set
thf(fact_126_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( member_nat2 @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_127_basic__trans__rules_I26_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_128_basic__trans__rules_I26_J,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_129_basic__trans__rules_I25_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_130_basic__trans__rules_I25_J,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_131_basic__trans__rules_I24_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% basic_trans_rules(24)
thf(fact_132_basic__trans__rules_I24_J,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% basic_trans_rules(24)
thf(fact_133_basic__trans__rules_I23_J,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% basic_trans_rules(23)
thf(fact_134_basic__trans__rules_I23_J,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% basic_trans_rules(23)
thf(fact_135_basic__trans__rules_I10_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_136_basic__trans__rules_I10_J,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_137_basic__trans__rules_I10_J,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_138_basic__trans__rules_I10_J,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_139_basic__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_140_basic__trans__rules_I9_J,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_141_basic__trans__rules_I9_J,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_142_basic__trans__rules_I9_J,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_143_basic__trans__rules_I8_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_144_basic__trans__rules_I8_J,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_145_basic__trans__rules_I8_J,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_146_basic__trans__rules_I8_J,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_147_basic__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_148_basic__trans__rules_I7_J,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_149_basic__trans__rules_I7_J,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_150_basic__trans__rules_I7_J,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_151_linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linear
thf(fact_152_linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linear
thf(fact_153_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_154_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_155_eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_156_eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_157_le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_158_le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_159_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_160_le__cases3,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_161_antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_162_antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_163_order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.eq_iff
thf(fact_164_order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% order.eq_iff
thf(fact_165_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_166_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_167_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_168_order__antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_169_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_170_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_171_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_172_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A5: real,B3: real] :
( ( ord_less_eq_real @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: real,B3: real] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_173_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_174_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_175_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_176_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_177_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_178_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_179_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_180_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_181_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_182_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_183_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_184_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_185_finite__has__minimal2,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( member8440522571783428010at_nat @ A2 @ A )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
& ( ord_le8460144461188290721at_nat @ X3 @ A2 )
& ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ A )
=> ( ( ord_le8460144461188290721at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_186_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_187_finite__has__minimal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real2 @ A2 @ A )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A )
& ( ord_less_eq_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_188_finite__has__maximal2,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( member8440522571783428010at_nat @ A2 @ A )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
& ( ord_le8460144461188290721at_nat @ A2 @ X3 )
& ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ A )
=> ( ( ord_le8460144461188290721at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_189_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_190_finite__has__maximal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real2 @ A2 @ A )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A )
& ( ord_less_eq_real @ A2 @ X3 )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_191_sorted__list__of__set__inject,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B4 ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( A = B4 ) ) ) ) ).
% sorted_list_of_set_inject
thf(fact_192_le__bot,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% le_bot
thf(fact_193_le__bot,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% le_bot
thf(fact_194_le__bot,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
=> ( A2 = bot_bot_set_real ) ) ).
% le_bot
thf(fact_195_le__bot,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% le_bot
thf(fact_196_bot__least,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot_least
thf(fact_197_bot__least,axiom,
! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% bot_least
thf(fact_198_bot__least,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% bot_least
thf(fact_199_bot__least,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot_least
thf(fact_200_bot__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot_unique
thf(fact_201_bot__unique,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% bot_unique
thf(fact_202_bot__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% bot_unique
thf(fact_203_bot__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot_unique
thf(fact_204_finite__has__minimal,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( A != bot_bo2099793752762293965at_nat )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
& ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ A )
=> ( ( ord_le8460144461188290721at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_205_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_206_finite__has__minimal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_207_finite__has__maximal,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( A != bot_bo2099793752762293965at_nat )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
& ! [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ A )
=> ( ( ord_le8460144461188290721at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_208_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_209_finite__has__maximal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_210_card__le__if__inj__on__rel,axiom,
! [B4: set_real,A: set_real,R: real > real > $o] :
( ( finite_finite_real @ B4 )
=> ( ! [A5: real] :
( ( member_real2 @ A5 @ A )
=> ? [B5: real] :
( ( member_real2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: real,A22: real,B3: real] :
( ( member_real2 @ A1 @ A )
=> ( ( member_real2 @ A22 @ A )
=> ( ( member_real2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A ) @ ( finite_card_real @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_211_card__le__if__inj__on__rel,axiom,
! [B4: set_real,A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > real > $o] :
( ( finite_finite_real @ B4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ A )
=> ? [B5: real] :
( ( member_real2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,B3: real] :
( ( member8440522571783428010at_nat @ A1 @ A )
=> ( ( member8440522571783428010at_nat @ A22 @ A )
=> ( ( member_real2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite_card_real @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_212_card__le__if__inj__on__rel,axiom,
! [B4: set_real,A: set_nat,R: nat > real > $o] :
( ( finite_finite_real @ B4 )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ? [B5: real] :
( ( member_real2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B3: real] :
( ( member_nat2 @ A1 @ A )
=> ( ( member_nat2 @ A22 @ A )
=> ( ( member_real2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_real @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_213_card__le__if__inj__on__rel,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_real,R: real > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ! [A5: real] :
( ( member_real2 @ A5 @ A )
=> ? [B5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: real,A22: real,B3: product_prod_nat_nat] :
( ( member_real2 @ A1 @ A )
=> ( ( member_real2 @ A22 @ A )
=> ( ( member8440522571783428010at_nat @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_214_card__le__if__inj__on__rel,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ A )
=> ? [B5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A1 @ A )
=> ( ( member8440522571783428010at_nat @ A22 @ A )
=> ( ( member8440522571783428010at_nat @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_215_card__le__if__inj__on__rel,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_nat,R: nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ? [B5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B3: product_prod_nat_nat] :
( ( member_nat2 @ A1 @ A )
=> ( ( member_nat2 @ A22 @ A )
=> ( ( member8440522571783428010at_nat @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_216_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A: set_real,R: real > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A5: real] :
( ( member_real2 @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: real,A22: real,B3: nat] :
( ( member_real2 @ A1 @ A )
=> ( ( member_real2 @ A22 @ A )
=> ( ( member_nat2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_217_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,B3: nat] :
( ( member8440522571783428010at_nat @ A1 @ A )
=> ( ( member8440522571783428010at_nat @ A22 @ A )
=> ( ( member_nat2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_218_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat2 @ B5 @ B4 )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat2 @ A1 @ A )
=> ( ( member_nat2 @ A22 @ A )
=> ( ( member_nat2 @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_219_finite__indexed__bound,axiom,
! [A: set_real,P: real > nat > $o] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M3: nat] :
! [X4: real] :
( ( member_real2 @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_220_finite__indexed__bound,axiom,
! [A: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M3: nat] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_221_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M3: nat] :
! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_222_finite__indexed__bound,axiom,
! [A: set_real,P: real > real > $o] :
( ( finite_finite_real @ A )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M3: real] :
! [X4: real] :
( ( member_real2 @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_223_finite__indexed__bound,axiom,
! [A: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > real > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M3: real] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_224_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M3: real] :
! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M3 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_225_enumerate__mono__le__iff,axiom,
! [S: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_226_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_227_card__0__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ) ).
% card_0_eq
thf(fact_228_card__0__eq,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( ( finite_card_real @ A )
= zero_zero_nat )
= ( A = bot_bot_set_real ) ) ) ).
% card_0_eq
thf(fact_229_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_230_arg__min__least,axiom,
! [S: set_Pr1261947904930325089at_nat,Y2: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ( ( member8440522571783428010at_nat @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic4984276347100956536at_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_231_arg__min__least,axiom,
! [S: set_real,Y2: real,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real2 @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_232_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y2 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_233_arg__min__least,axiom,
! [S: set_Pr1261947904930325089at_nat,Y2: product_prod_nat_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ( ( member8440522571783428010at_nat @ Y2 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic7428442014618555988t_real @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_234_arg__min__least,axiom,
! [S: set_real,Y2: real,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real2 @ Y2 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_235_class__dense__linordered__field_Osorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_real] :
( ~ ( finite_finite_real @ A )
=> ( ( sorted6366500744023230182t_real @ ord_less_eq_real @ A )
= nil_real ) ) ).
% class_dense_linordered_field.sorted_list_of_set.fold_insort_key.infinite
thf(fact_236_class__dense__linordered__field_Osorted__list__of__set_Ofold__insort__key_Oempty,axiom,
( ( sorted6366500744023230182t_real @ ord_less_eq_real @ bot_bot_set_real )
= nil_real ) ).
% class_dense_linordered_field.sorted_list_of_set.fold_insort_key.empty
thf(fact_237_GreatestI2__order,axiom,
! [P: real > $o,X2: real,Q: real > $o] :
( ( P @ X2 )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y4: real] :
( ( P @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_238_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_239_Greatest__equality,axiom,
! [P: real > $o,X2: real] :
( ( P @ X2 )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( order_Greatest_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_240_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_241_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_242_card__eq__0__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bo2099793752762293965at_nat )
| ~ ( finite6177210948735845034at_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_243_card__eq__0__iff,axiom,
! [A: set_real] :
( ( ( finite_card_real @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_real )
| ~ ( finite_finite_real @ A ) ) ) ).
% card_eq_0_iff
thf(fact_244_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_245_empty__subsetI,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% empty_subsetI
thf(fact_246_empty__subsetI,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% empty_subsetI
thf(fact_247_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_248_subset__empty,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% subset_empty
thf(fact_249_subset__empty,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_250_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_251_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_252_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_253_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_254_card_Oempty,axiom,
( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_255_card_Oempty,axiom,
( ( finite_card_real @ bot_bot_set_real )
= zero_zero_nat ) ).
% card.empty
thf(fact_256_card_Oinfinite,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A )
=> ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_257_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_258_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_259_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_260_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat2 @ S2 @ S )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S @ N3 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_261_less__eq__list__code_I2_J,axiom,
! [Xs: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ Xs ) ).
% less_eq_list_code(2)
thf(fact_262_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_263_finite__subset,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( finite6177210948735845034at_nat @ B4 )
=> ( finite6177210948735845034at_nat @ A ) ) ) ).
% finite_subset
thf(fact_264_finite__subset,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_265_infinite__super,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ S @ T )
=> ( ~ ( finite6177210948735845034at_nat @ S )
=> ~ ( finite6177210948735845034at_nat @ T ) ) ) ).
% infinite_super
thf(fact_266_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_267_rev__finite__subset,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( finite6177210948735845034at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_268_rev__finite__subset,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_269_subset__code_I1_J,axiom,
! [Xs: list_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
= ( ! [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( member_real2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_270_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B4 )
= ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_271_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_272_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_273_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_274_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_275_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_276_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_277_enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).
% enumerate_in_set
thf(fact_278_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_279_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_280_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_281_card__subset__eq,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( ( finite711546835091564841at_nat @ A )
= ( finite711546835091564841at_nat @ B4 ) )
=> ( A = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_282_card__subset__eq,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B4 ) )
=> ( A = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_283_infinite__arbitrarily__large,axiom,
! [A: set_Pr1261947904930325089at_nat,N: nat] :
( ~ ( finite6177210948735845034at_nat @ A )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B6 )
& ( ( finite711546835091564841at_nat @ B6 )
= N )
& ( ord_le3146513528884898305at_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_284_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_285_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_286_set__nths__subset,axiom,
! [Xs: list_nat,I: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_287_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_288_card__mono,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_289_card__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_290_card__seteq,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B4 ) @ ( finite711546835091564841at_nat @ A ) )
=> ( A = B4 ) ) ) ) ).
% card_seteq
thf(fact_291_card__seteq,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A ) )
=> ( A = B4 ) ) ) ) ).
% card_seteq
thf(fact_292_exists__subset__between,axiom,
! [A: set_Pr1261947904930325089at_nat,N: nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ C2 ) )
=> ( ( ord_le3146513528884898305at_nat @ A @ C2 )
=> ( ( finite6177210948735845034at_nat @ C2 )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ C2 )
& ( ( finite711546835091564841at_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_293_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_294_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S ) )
=> ~ ! [T2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ T2 @ S )
=> ( ( ( finite711546835091564841at_nat @ T2 )
= N )
=> ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_295_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_296_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
( ! [G: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ G @ F2 )
=> ( ( finite6177210948735845034at_nat @ G )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G ) @ C2 ) ) )
=> ( ( finite6177210948735845034at_nat @ F2 )
& ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_297_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_298_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M4: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_299_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat2 @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_300_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_301_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_302_ex__card,axiom,
! [N: nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ A ) )
=> ? [S3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ S3 @ A )
& ( ( finite711546835091564841at_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_303_ex__card,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A )
& ( ( finite_card_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_304_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_305_rel__simps_I46_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% rel_simps(46)
thf(fact_306_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( member_nat2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_307_subsetI,axiom,
! [A: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real2 @ X3 @ A )
=> ( member_real2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A @ B4 ) ) ).
% subsetI
thf(fact_308_subsetI,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
=> ( member8440522571783428010at_nat @ X3 @ B4 ) )
=> ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_309_Set_Obasic__monos_I7_J,axiom,
! [A: set_nat,B4: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ X2 @ B4 ) ) ) ).
% Set.basic_monos(7)
thf(fact_310_Set_Obasic__monos_I7_J,axiom,
! [A: set_real,B4: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ( member_real2 @ X2 @ A )
=> ( member_real2 @ X2 @ B4 ) ) ) ).
% Set.basic_monos(7)
thf(fact_311_Set_Obasic__monos_I7_J,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( member8440522571783428010at_nat @ X2 @ B4 ) ) ) ).
% Set.basic_monos(7)
thf(fact_312_Set_Obasic__monos_I6_J,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_313_Set_Obasic__monos_I6_J,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_314_basic__trans__rules_I31_J,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B4 ) ) ) ).
% basic_trans_rules(31)
thf(fact_315_basic__trans__rules_I31_J,axiom,
! [A: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B4 )
=> ( ( member_real2 @ C @ A )
=> ( member_real2 @ C @ B4 ) ) ) ).
% basic_trans_rules(31)
thf(fact_316_basic__trans__rules_I31_J,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( member8440522571783428010at_nat @ C @ A )
=> ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% basic_trans_rules(31)
thf(fact_317_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( member_nat2 @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_318_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B7: set_real] :
! [X: real] :
( ( member_real2 @ X @ A3 )
=> ( member_real2 @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_319_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A3 )
=> ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_320_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
! [T3: nat] :
( ( member_nat2 @ T3 @ A3 )
=> ( member_nat2 @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_321_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B7: set_real] :
! [T3: real] :
( ( member_real2 @ T3 @ A3 )
=> ( member_real2 @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_322_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A3 )
=> ( member8440522571783428010at_nat @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_323_Collect__mono__iff,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
= ( ! [X: product_prod_nat_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_324_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_325_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_326_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_327_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_328_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_329_less__by__empty,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( A = bot_bo2099793752762293965at_nat )
=> ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ).
% less_by_empty
thf(fact_330_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_331_subset__emptyI,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A )
=> ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_332_subset__emptyI,axiom,
! [A: set_real] :
( ! [X3: real] :
~ ( member_real2 @ X3 @ A )
=> ( ord_less_eq_set_real @ A @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_333_finite__enum__subset,axiom,
! [X5: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X5 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_334_zero__order_I5_J,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_335_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_336_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_337_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_338_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_339_enumerate__mono__iff,axiom,
! [S: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_340_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M2: nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_341_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_342_semiring__norm_I137_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% semiring_norm(137)
thf(fact_343_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_344_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_345_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_346_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_347_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_348_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_349_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_350_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_351_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_352_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_353_order__less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_354_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_355_order__less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_356_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_357_order__less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_358_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_359_linorder__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_360_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_361_order__less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_362_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_363_order__less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_364_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_365_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_366_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_367_order__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_368_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_369_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_370_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_371_order__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_372_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_373_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_374_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_375_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_376_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_377_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_378_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_379_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_380_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_381_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_382_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_383_order__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_384_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_385_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_386_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_387_linorder__neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_388_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_389_order__less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_390_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_391_linorder__neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_392_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_393_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_394_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_395_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_396_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_397_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_398_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_399_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_400_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_401_order_Ostrict__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_402_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_403_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A5: real,B3: real] :
( ( ord_less_real @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B3: real] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_404_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P2 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_405_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_406_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_407_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_408_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_409_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_410_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_411_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_412_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_413_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_414_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_415_ord__less__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_416_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_417_ord__eq__less__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_418_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_419_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_420_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_421_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_422_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z: real] :
( ( ord_less_real @ X2 @ Z )
& ( ord_less_real @ Z @ Y2 ) ) ) ).
% dense
thf(fact_423_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_424_gt__ex,axiom,
! [X2: real] :
? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% gt_ex
thf(fact_425_lt__ex,axiom,
! [X2: real] :
? [Y: real] : ( ord_less_real @ Y @ X2 ) ).
% lt_ex
thf(fact_426_class__dense__linordered__field_Olt__ex,axiom,
! [X2: real] :
? [Y: real] : ( ord_less_real @ Y @ X2 ) ).
% class_dense_linordered_field.lt_ex
thf(fact_427_class__dense__linordered__field_Ogt__ex,axiom,
! [X2: real] :
? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% class_dense_linordered_field.gt_ex
thf(fact_428_enumerate__mono,axiom,
! [M2: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% enumerate_mono
thf(fact_429_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_430_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_431_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_432_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_433_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_434_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_435_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_436_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_437_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_438_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_439_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_440_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_441_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_442_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_443_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_444_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_445_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_446_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_447_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_448_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_449_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_450_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_451_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_452_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_453_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_454_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_455_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_456_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_457_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_458_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_459_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_460_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_461_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_462_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_463_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_464_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ( X != Y3 ) ) ) ) ).
% order_less_le
thf(fact_465_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_466_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_less
thf(fact_467_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_468_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_469_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_470_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_471_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_472_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_473_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_474_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_475_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_476_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_477_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_478_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_479_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_480_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_481_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_482_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_483_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_484_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_485_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_486_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_487_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_488_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_489_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_490_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_491_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_492_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_493_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_494_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_495_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_496_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_497_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_498_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_499_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_500_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_501_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_502_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_503_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_504_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_505_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_506_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_507_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_508_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_509_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_510_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_511_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_512_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_513_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_514_bot_Onot__eq__extremum,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( A2 != bot_bo2099793752762293965at_nat )
= ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_515_bot_Onot__eq__extremum,axiom,
! [A2: set_real] :
( ( A2 != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_516_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_517_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_518_bot_Oextremum__strict,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% bot.extremum_strict
thf(fact_519_bot_Oextremum__strict,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_520_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_521_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_522_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_523_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_524_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_525_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_526_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_527_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_528_le__simps_I1_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% le_simps(1)
thf(fact_529_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_530_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_531_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_532_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_533_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_534_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_535_finite__maxlen,axiom,
! [M4: set_list_nat] :
( ( finite8100373058378681591st_nat @ M4 )
=> ? [N3: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M4 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_536_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
& ( member_nat2 @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_537_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_538_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( member_nat2 @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_539_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat2 @ X @ N4 )
=> ( ord_less_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_540_finite__enumerate__mono,axiom,
! [M2: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M2 ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_541_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ X5 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_542_infinite__growing,axiom,
! [X5: set_real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ X5 )
=> ? [Xa: real] :
( ( member_real2 @ Xa @ X5 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X5 ) ) ) ).
% infinite_growing
thf(fact_543_ex__min__if__finite,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S )
& ~ ? [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ S )
& ( ord_le1203424502768444845at_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_544_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat2 @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_545_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real2 @ X3 @ S )
& ~ ? [Xa: real] :
( ( member_real2 @ Xa @ S )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_546_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_547_nths__all,axiom,
! [Xs: list_nat,I: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ I3 @ I ) )
=> ( ( nths_nat @ Xs @ I )
= Xs ) ) ).
% nths_all
thf(fact_548_card__ge__0__finite,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A ) )
=> ( finite6177210948735845034at_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_549_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_550_length__pos__if__in__set,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_551_length__pos__if__in__set,axiom,
! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_552_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_553_finite__enum__ext,axiom,
! [X5: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X5 ) )
=> ( ( infini8530281810654367211te_nat @ X5 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X5 )
= ( finite_card_nat @ Y6 ) )
=> ( X5 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_554_finite__enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat2 @ S2 @ S )
=> ? [N3: nat] :
( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N3 )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_555_finite__enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_556_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_557_arg__min__if__finite_I2_J,axiom,
! [S: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ~ ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic4984276347100956536at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_558_arg__min__if__finite_I2_J,axiom,
! [S: set_real,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ~ ? [X4: real] :
( ( member_real2 @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic5055836439445974935al_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_559_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_560_arg__min__if__finite_I2_J,axiom,
! [S: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ~ ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ S )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic7428442014618555988t_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_561_arg__min__if__finite_I2_J,axiom,
! [S: set_real,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ~ ? [X4: real] :
( ( member_real2 @ X4 @ S )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_562_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_563_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_564_card__gt__0__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A ) )
= ( ( A != bot_bo2099793752762293965at_nat )
& ( finite6177210948735845034at_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_565_card__gt__0__iff,axiom,
! [A: set_real] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A ) )
= ( ( A != bot_bot_set_real )
& ( finite_finite_real @ A ) ) ) ).
% card_gt_0_iff
thf(fact_566_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_567_verit__comp__simplify_I3_J,axiom,
! [B8: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A6 ) )
= ( ord_less_nat @ A6 @ B8 ) ) ).
% verit_comp_simplify(3)
thf(fact_568_verit__comp__simplify_I3_J,axiom,
! [B8: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B8 @ A6 ) )
= ( ord_less_real @ A6 @ B8 ) ) ).
% verit_comp_simplify(3)
thf(fact_569_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_570_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_571_not__psubset__empty,axiom,
! [A: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% not_psubset_empty
thf(fact_572_not__psubset__empty,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_573_finite__psubset__induct,axiom,
! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [A7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A7 )
=> ( ! [B9: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_574_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_575_not__less__Nil,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ nil_nat ) ).
% not_less_Nil
thf(fact_576_less__list__code_I1_J,axiom,
! [Xs: list_nat] :
~ ( ord_less_list_nat @ Xs @ nil_nat ) ).
% less_list_code(1)
thf(fact_577_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_578_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_579_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_580_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_581_psubset__card__mono,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le7866589430770878221at_nat @ A @ B4 )
=> ( ord_less_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_582_psubset__card__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_583_card__psubset,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) )
=> ( ord_le7866589430770878221at_nat @ A @ B4 ) ) ) ) ).
% card_psubset
thf(fact_584_card__psubset,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_set_nat @ A @ B4 ) ) ) ) ).
% card_psubset
thf(fact_585_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_586_complete__interval,axiom,
! [A2: real,B: real,P: real > $o] :
( ( ord_less_real @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: real] :
( ( ord_less_eq_real @ A2 @ C4 )
& ( ord_less_eq_real @ C4 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_real @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_587_sorted__list__of__set__unique,axiom,
! [A: set_nat,L: list_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A ) ) )
= ( ( linord2614967742042102400et_nat @ A )
= L ) ) ) ).
% sorted_list_of_set_unique
thf(fact_588_sorted__list__of__set__unique,axiom,
! [A: set_real,L: list_real] :
( ( finite_finite_real @ A )
=> ( ( ( sorted_wrt_real @ ord_less_real @ L )
& ( ( set_real2 @ L )
= A )
& ( ( size_size_list_real @ L )
= ( finite_card_real @ A ) ) )
= ( ( linord4252657396651189596t_real @ A )
= L ) ) ) ).
% sorted_list_of_set_unique
thf(fact_589_finite__enumerate__step,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_590_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_591_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_592_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_593_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_594_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_595_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_596_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_597_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_598_psubsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_599_psubsetD,axiom,
! [A: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A @ B4 )
=> ( ( member_real2 @ C @ A )
=> ( member_real2 @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_600_psubsetD,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A @ B4 )
=> ( ( member8440522571783428010at_nat @ C @ A )
=> ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_601_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_602_sorted__wrt__mono__rel,axiom,
! [Xs: list_real,P: real > real > $o,Q: real > real > $o] :
( ! [X3: real,Y: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( member_real2 @ Y @ ( set_real2 @ Xs ) )
=> ( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) ) ) )
=> ( ( sorted_wrt_real @ P @ Xs )
=> ( sorted_wrt_real @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_603_sorted__wrt__mono__rel,axiom,
! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) ) ) )
=> ( ( sorted5214655850825725294at_nat @ P @ Xs )
=> ( sorted5214655850825725294at_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_604_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_605_nat_Osimps_I3_J,axiom,
! [X22: nat] :
( ( suc @ X22 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_606_old_Onat_Osimps_I3_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_607_old_Onat_Osimps_I2_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.simps(2)
thf(fact_608_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_609_nat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [X23: nat] :
( Y2
!= ( suc @ X23 ) ) ) ).
% nat.exhaust
thf(fact_610_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_611_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_612_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_613_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_614_Suc__not__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_615_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_616_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_617_not__less__simps_I1_J,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_simps(1)
thf(fact_618_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_619_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_620_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_621_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_622_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_623_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_624_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_625_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_626_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_627_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_628_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_629_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_630_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_631_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_632_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_633_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_634_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z: nat] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_635_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_636_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_637_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_638_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_639_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_640_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_641_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_642_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_643_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_644_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_645_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_646_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_647_strict__sorted__imp__sorted,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_648_sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted_simps(1)
thf(fact_649_sorted__simps_I1_J,axiom,
sorted_wrt_real @ ord_less_eq_real @ nil_real ).
% sorted_simps(1)
thf(fact_650_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_651_strict__sorted__simps_I1_J,axiom,
sorted_wrt_real @ ord_less_real @ nil_real ).
% strict_sorted_simps(1)
thf(fact_652_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_653_strict__sorted__equal,axiom,
! [Xs: list_real,Ys: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
=> ( ( sorted_wrt_real @ ord_less_real @ Ys )
=> ( ( ( set_real2 @ Ys )
= ( set_real2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_654_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_655_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_real] : ( sorted_wrt_real @ ord_less_eq_real @ ( linord4252657396651189596t_real @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_656_strict__sorted__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% strict_sorted_list_of_set
thf(fact_657_strict__sorted__list__of__set,axiom,
! [A: set_real] : ( sorted_wrt_real @ ord_less_real @ ( linord4252657396651189596t_real @ A ) ) ).
% strict_sorted_list_of_set
thf(fact_658_sorted__nths,axiom,
! [Xs: list_nat,I: set_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( nths_nat @ Xs @ I ) ) ) ).
% sorted_nths
thf(fact_659_sorted__nths,axiom,
! [Xs: list_real,I: set_nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( nths_real @ Xs @ I ) ) ) ).
% sorted_nths
thf(fact_660_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_661_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_662_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_663_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_real @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_664_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_665_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_666_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_667_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_668_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_669_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_670_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_671_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_672_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_673_le__simps_I3_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% le_simps(3)
thf(fact_674_le__simps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% le_simps(2)
thf(fact_675_not__less__simps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_simps(2)
thf(fact_676_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_677_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_678_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_679_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_680_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_681_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_682_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_683_enumerate__step,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).
% enumerate_step
thf(fact_684_card__le__Suc0__iff__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A )
=> ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ A )
=> ( X = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_685_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ A )
=> ( X = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_686_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_687_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ~ ! [L2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ L2 )
=> ( ( ( set_real2 @ L2 )
= A )
=> ( ( size_size_list_real @ L2 )
!= ( finite_card_real @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_688_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_689_encode__unary__nat_Ocases,axiom,
! [X2: nat] :
( ! [L2: nat] :
( X2
!= ( suc @ L2 ) )
=> ( X2 = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_690_fib_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ( ( X2
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ ( suc @ N3 ) ) ) ) ) ).
% fib.cases
thf(fact_691_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_692_card__set__1__iff__replicate,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( finite711546835091564841at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_Pr5478986624290739719at_nat )
& ? [X: product_prod_nat_nat] :
( Xs
= ( replic4235873036481779905at_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ X ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_693_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_694_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_695_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_696_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_697_Ball__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
=> ( P @ X ) ) )
= ( ( P @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_698_Bex__set__replicate,axiom,
! [N: nat,A2: nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
& ( P @ X ) ) )
= ( ( P @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_699_in__set__replicate,axiom,
! [X2: real,N: nat,Y2: real] :
( ( member_real2 @ X2 @ ( set_real2 @ ( replicate_real @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_700_in__set__replicate,axiom,
! [X2: product_prod_nat_nat,N: nat,Y2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_701_in__set__replicate,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_702_replicate_Osimps_I1_J,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate.simps(1)
thf(fact_703_replicate__length__same,axiom,
! [Xs: list_nat,X2: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_704_replicate__eqI,axiom,
! [Xs: list_real,N: nat,X2: real] :
( ( ( size_size_list_real @ Xs )
= N )
=> ( ! [Y: real] :
( ( member_real2 @ Y @ ( set_real2 @ Xs ) )
=> ( Y = X2 ) )
=> ( Xs
= ( replicate_real @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_705_replicate__eqI,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat,X2: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= N )
=> ( ! [Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( Y = X2 ) )
=> ( Xs
= ( replic4235873036481779905at_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_706_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( Y = X2 ) )
=> ( Xs
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_707_sorted__replicate,axiom,
! [N: nat,X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_708_sorted__replicate,axiom,
! [N: nat,X2: real] : ( sorted_wrt_real @ ord_less_eq_real @ ( replicate_real @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_709_set__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_710_set__replicate,axiom,
! [N: nat,X2: product_prod_nat_nat] :
( ( N != zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X2 ) )
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% set_replicate
thf(fact_711_set__replicate,axiom,
! [N: nat,X2: real] :
( ( N != zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X2 ) )
= ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ).
% set_replicate
thf(fact_712_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_713_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_714_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B @ A ) )
= ( ( A2 = B )
| ( member_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_715_insert__iff,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real2 @ A2 @ ( insert_real2 @ B @ A ) )
= ( ( A2 = B )
| ( member_real2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_716_insert__iff,axiom,
! [A2: product_prod_nat_nat,B: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ A ) )
= ( ( A2 = B )
| ( member8440522571783428010at_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_717_insertCI,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat2 @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B @ B4 ) ) ) ).
% insertCI
thf(fact_718_insertCI,axiom,
! [A2: real,B4: set_real,B: real] :
( ( ~ ( member_real2 @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_real2 @ A2 @ ( insert_real2 @ B @ B4 ) ) ) ).
% insertCI
thf(fact_719_insertCI,axiom,
! [A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_720_insert__subset,axiom,
! [X2: nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ A ) @ B4 )
= ( ( member_nat2 @ X2 @ B4 )
& ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_721_insert__subset,axiom,
! [X2: real,A: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( insert_real2 @ X2 @ A ) @ B4 )
= ( ( member_real2 @ X2 @ B4 )
& ( ord_less_eq_set_real @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_722_insert__subset,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) @ B4 )
= ( ( member8440522571783428010at_nat @ X2 @ B4 )
& ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_723_singletonI,axiom,
! [A2: nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_724_singletonI,axiom,
! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_725_singletonI,axiom,
! [A2: real] : ( member_real2 @ A2 @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_726_finite__insert,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) )
= ( finite6177210948735845034at_nat @ A ) ) ).
% finite_insert
thf(fact_727_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_728_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_729_remdups__adj__set,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( remdups_adj_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_730_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_731_singleton__insert__inj__eq,axiom,
! [B: product_prod_nat_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_732_singleton__insert__inj__eq,axiom,
! [B: real,A2: real,A: set_real] :
( ( ( insert_real2 @ B @ bot_bot_set_real )
= ( insert_real2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_real @ A @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_733_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_734_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A2 @ A )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( ( A2 = B )
& ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_735_singleton__insert__inj__eq_H,axiom,
! [A2: real,A: set_real,B: real] :
( ( ( insert_real2 @ A2 @ A )
= ( insert_real2 @ B @ bot_bot_set_real ) )
= ( ( A2 = B )
& ( ord_less_eq_set_real @ A @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_736_List_Oset__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X2 @ Xs ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_737_card__insert__disjoint,axiom,
! [A: set_real,X2: real] :
( ( finite_finite_real @ A )
=> ( ~ ( member_real2 @ X2 @ A )
=> ( ( finite_card_real @ ( insert_real2 @ X2 @ A ) )
= ( suc @ ( finite_card_real @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_738_card__insert__disjoint,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) )
= ( suc @ ( finite711546835091564841at_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_739_card__insert__disjoint,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat2 @ X2 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_740_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_741_subset__insert,axiom,
! [X2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_742_subset__insert,axiom,
! [X2: real,A: set_real,B4: set_real] :
( ~ ( member_real2 @ X2 @ A )
=> ( ( ord_less_eq_set_real @ A @ ( insert_real2 @ X2 @ B4 ) )
= ( ord_less_eq_set_real @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_743_subset__insert,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ B4 ) )
= ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_744_insert__subsetI,axiom,
! [X2: nat,A: set_nat,X5: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_745_insert__subsetI,axiom,
! [X2: real,A: set_real,X5: set_real] :
( ( member_real2 @ X2 @ A )
=> ( ( ord_less_eq_set_real @ X5 @ A )
=> ( ord_less_eq_set_real @ ( insert_real2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_746_insert__subsetI,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( ord_le3146513528884898305at_nat @ X5 @ A )
=> ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_747_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_748_singletonD,axiom,
! [B: product_prod_nat_nat,A2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_749_singletonD,axiom,
! [B: real,A2: real] :
( ( member_real2 @ B @ ( insert_real2 @ A2 @ bot_bot_set_real ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_750_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_751_singleton__iff,axiom,
! [B: product_prod_nat_nat,A2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_752_singleton__iff,axiom,
! [B: real,A2: real] :
( ( member_real2 @ B @ ( insert_real2 @ A2 @ bot_bot_set_real ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_753_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D3 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D3 ) )
| ( ( A2 = D3 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_754_doubleton__eq__iff,axiom,
! [A2: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D3: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D3 @ bot_bo2099793752762293965at_nat ) ) )
= ( ( ( A2 = C )
& ( B = D3 ) )
| ( ( A2 = D3 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_755_doubleton__eq__iff,axiom,
! [A2: real,B: real,C: real,D3: real] :
( ( ( insert_real2 @ A2 @ ( insert_real2 @ B @ bot_bot_set_real ) )
= ( insert_real2 @ C @ ( insert_real2 @ D3 @ bot_bot_set_real ) ) )
= ( ( ( A2 = C )
& ( B = D3 ) )
| ( ( A2 = D3 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_756_empty__not__insert,axiom,
! [A2: nat,A: set_nat] :
( bot_bot_set_nat
!= ( insert_nat2 @ A2 @ A ) ) ).
% empty_not_insert
thf(fact_757_empty__not__insert,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( bot_bo2099793752762293965at_nat
!= ( insert8211810215607154385at_nat @ A2 @ A ) ) ).
% empty_not_insert
thf(fact_758_empty__not__insert,axiom,
! [A2: real,A: set_real] :
( bot_bot_set_real
!= ( insert_real2 @ A2 @ A ) ) ).
% empty_not_insert
thf(fact_759_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_760_singleton__inject,axiom,
! [A2: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_761_singleton__inject,axiom,
! [A2: real,B: real] :
( ( ( insert_real2 @ A2 @ bot_bot_set_real )
= ( insert_real2 @ B @ bot_bot_set_real ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_762_finite_OinsertI,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_763_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_764_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B6 ) )
& ~ ( member_nat2 @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_765_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real2 @ A2 @ A )
=> ? [B6: set_real] :
( ( A
= ( insert_real2 @ A2 @ B6 ) )
& ~ ( member_real2 @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_766_mk__disjoint__insert,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A2 @ A )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ A2 @ B6 ) )
& ~ ( member8440522571783428010at_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_767_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ~ ( member_nat2 @ B @ B4 )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C5: set_nat] :
( ( A
= ( insert_nat2 @ B @ C5 ) )
& ~ ( member_nat2 @ B @ C5 )
& ( B4
= ( insert_nat2 @ A2 @ C5 ) )
& ~ ( member_nat2 @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_768_insert__eq__iff,axiom,
! [A2: real,A: set_real,B: real,B4: set_real] :
( ~ ( member_real2 @ A2 @ A )
=> ( ~ ( member_real2 @ B @ B4 )
=> ( ( ( insert_real2 @ A2 @ A )
= ( insert_real2 @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C5: set_real] :
( ( A
= ( insert_real2 @ B @ C5 ) )
& ~ ( member_real2 @ B @ C5 )
& ( B4
= ( insert_real2 @ A2 @ C5 ) )
& ~ ( member_real2 @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_769_insert__eq__iff,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A2 @ A )
=> ( ~ ( member8440522571783428010at_nat @ B @ B4 )
=> ( ( ( insert8211810215607154385at_nat @ A2 @ A )
= ( insert8211810215607154385at_nat @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C5: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ B @ C5 ) )
& ~ ( member8440522571783428010at_nat @ B @ C5 )
& ( B4
= ( insert8211810215607154385at_nat @ A2 @ C5 ) )
& ~ ( member8440522571783428010at_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_770_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_771_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real2 @ A2 @ A )
=> ( ( insert_real2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_772_insert__absorb,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A2 @ A )
=> ( ( insert8211810215607154385at_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_773_insert__ident,axiom,
! [X2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ~ ( member_nat2 @ X2 @ B4 )
=> ( ( ( insert_nat2 @ X2 @ A )
= ( insert_nat2 @ X2 @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_774_insert__ident,axiom,
! [X2: real,A: set_real,B4: set_real] :
( ~ ( member_real2 @ X2 @ A )
=> ( ~ ( member_real2 @ X2 @ B4 )
=> ( ( ( insert_real2 @ X2 @ A )
= ( insert_real2 @ X2 @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_775_insert__ident,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ( ( insert8211810215607154385at_nat @ X2 @ A )
= ( insert8211810215607154385at_nat @ X2 @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_776_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat2 @ X2 @ B6 ) )
=> ( member_nat2 @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_777_Set_Oset__insert,axiom,
! [X2: real,A: set_real] :
( ( member_real2 @ X2 @ A )
=> ~ ! [B6: set_real] :
( ( A
= ( insert_real2 @ X2 @ B6 ) )
=> ( member_real2 @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_778_Set_Oset__insert,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ A )
=> ~ ! [B6: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ X2 @ B6 ) )
=> ( member8440522571783428010at_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_779_insertI2,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( member_nat2 @ A2 @ B4 )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B @ B4 ) ) ) ).
% insertI2
thf(fact_780_insertI2,axiom,
! [A2: real,B4: set_real,B: real] :
( ( member_real2 @ A2 @ B4 )
=> ( member_real2 @ A2 @ ( insert_real2 @ B @ B4 ) ) ) ).
% insertI2
thf(fact_781_insertI2,axiom,
! [A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A2 @ B4 )
=> ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_782_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ B4 ) ) ).
% insertI1
thf(fact_783_insertI1,axiom,
! [A2: real,B4: set_real] : ( member_real2 @ A2 @ ( insert_real2 @ A2 @ B4 ) ) ).
% insertI1
thf(fact_784_insertI1,axiom,
! [A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_785_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_786_insertE,axiom,
! [A2: real,B: real,A: set_real] :
( ( member_real2 @ A2 @ ( insert_real2 @ B @ A ) )
=> ( ( A2 != B )
=> ( member_real2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_787_insertE,axiom,
! [A2: product_prod_nat_nat,B: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member8440522571783428010at_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_788_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_789_subset__singletonD,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_790_subset__singletonD,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
=> ( ( A = bot_bo2099793752762293965at_nat )
| ( A
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_791_subset__singletonD,axiom,
! [A: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) )
=> ( ( A = bot_bot_set_real )
| ( A
= ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_792_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_793_subset__singleton__iff,axiom,
! [X5: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ X5 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
= ( ( X5 = bot_bo2099793752762293965at_nat )
| ( X5
= ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_794_subset__singleton__iff,axiom,
! [X5: set_real,A2: real] :
( ( ord_less_eq_set_real @ X5 @ ( insert_real2 @ A2 @ bot_bot_set_real ) )
= ( ( X5 = bot_bot_set_real )
| ( X5
= ( insert_real2 @ A2 @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_795_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A5: nat] :
( A2
= ( insert_nat2 @ A5 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_796_finite_Ocases,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( A2 != bot_bo2099793752762293965at_nat )
=> ~ ! [A7: set_Pr1261947904930325089at_nat] :
( ? [A5: product_prod_nat_nat] :
( A2
= ( insert8211810215607154385at_nat @ A5 @ A7 ) )
=> ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_797_finite_Ocases,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ~ ! [A7: set_real] :
( ? [A5: real] :
( A2
= ( insert_real2 @ A5 @ A7 ) )
=> ~ ( finite_finite_real @ A7 ) ) ) ) ).
% finite.cases
thf(fact_798_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B2: nat] :
( ( A4
= ( insert_nat2 @ B2 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_799_finite_Osimps,axiom,
( finite6177210948735845034at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
| ? [A3: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( A4
= ( insert8211810215607154385at_nat @ B2 @ A3 ) )
& ( finite6177210948735845034at_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_800_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A4: set_real] :
( ( A4 = bot_bot_set_real )
| ? [A3: set_real,B2: real] :
( ( A4
= ( insert_real2 @ B2 @ A3 ) )
& ( finite_finite_real @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_801_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_802_finite__induct,axiom,
! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F2 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ~ ( member8440522571783428010at_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8211810215607154385at_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_803_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_804_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_805_finite__ne__induct,axiom,
! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F2 )
=> ( ( F2 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
=> ( ! [X3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( F3 != bot_bo2099793752762293965at_nat )
=> ( ~ ( member8440522571783428010at_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8211810215607154385at_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_806_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X3: real] : ( P @ ( insert_real2 @ X3 @ bot_bot_set_real ) )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_807_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_808_infinite__finite__induct,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
( ! [A7: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ~ ( member8440522571783428010at_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8211810215607154385at_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_809_infinite__finite__induct,axiom,
! [P: set_real > $o,A: set_real] :
( ! [A7: set_real] :
( ~ ( finite_finite_real @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real2 @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_810_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_811_sorted__remdups__adj,axiom,
! [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( remdups_adj_real @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_812_card__insert__le,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) ) ) ).
% card_insert_le
thf(fact_813_card__insert__le,axiom,
! [A: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) ) ) ).
% card_insert_le
thf(fact_814_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat2 @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat2 @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_815_finite__ranking__induct,axiom,
! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,S3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ S3 )
=> ( ! [Y4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert8211810215607154385at_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_816_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ! [Y4: real] :
( ( member_real2 @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real2 @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_817_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat2 @ Y4 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat2 @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_818_finite__ranking__induct,axiom,
! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,S3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ S3 )
=> ( ! [Y4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y4 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert8211810215607154385at_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_819_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ! [Y4: real] :
( ( member_real2 @ Y4 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real2 @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_820_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A7 )
=> ( ord_less_nat @ X4 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat2 @ B3 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_821_finite__linorder__max__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ! [X4: real] :
( ( member_real2 @ X4 @ A7 )
=> ( ord_less_real @ X4 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_real2 @ B3 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_822_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A7 )
=> ( ord_less_nat @ B3 @ X4 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat2 @ B3 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_823_finite__linorder__min__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ! [X4: real] :
( ( member_real2 @ X4 @ A7 )
=> ( ord_less_real @ B3 @ X4 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_real2 @ B3 @ A7 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_824_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ~ ( member_nat2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_825_finite__subset__induct,axiom,
! [F2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F2 )
=> ( ( ord_le3146513528884898305at_nat @ F2 @ A )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( member8440522571783428010at_nat @ A5 @ A )
=> ( ~ ( member8440522571783428010at_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8211810215607154385at_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_826_finite__subset__induct,axiom,
! [F2: set_real,A: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real2 @ A5 @ A )
=> ( ~ ( member_real2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real2 @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_827_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_828_finite__subset__induct_H,axiom,
! [F2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F2 )
=> ( ( ord_le3146513528884898305at_nat @ F2 @ A )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( member8440522571783428010at_nat @ A5 @ A )
=> ( ( ord_le3146513528884898305at_nat @ F3 @ A )
=> ( ~ ( member8440522571783428010at_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8211810215607154385at_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_829_finite__subset__induct_H,axiom,
! [F2: set_real,A: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real2 @ A5 @ A )
=> ( ( ord_less_eq_set_real @ F3 @ A )
=> ( ~ ( member_real2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real2 @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_830_card__Suc__eq__finite,axiom,
! [A: set_real,K: nat] :
( ( ( finite_card_real @ A )
= ( suc @ K ) )
= ( ? [B2: real,B7: set_real] :
( ( A
= ( insert_real2 @ B2 @ B7 ) )
& ~ ( member_real2 @ B2 @ B7 )
& ( ( finite_card_real @ B7 )
= K )
& ( finite_finite_real @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_831_card__Suc__eq__finite,axiom,
! [A: set_Pr1261947904930325089at_nat,K: nat] :
( ( ( finite711546835091564841at_nat @ A )
= ( suc @ K ) )
= ( ? [B2: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ B2 @ B7 ) )
& ~ ( member8440522571783428010at_nat @ B2 @ B7 )
& ( ( finite711546835091564841at_nat @ B7 )
= K )
& ( finite6177210948735845034at_nat @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_832_card__Suc__eq__finite,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B2: nat,B7: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ B7 ) )
& ~ ( member_nat2 @ B2 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_833_card__insert__if,axiom,
! [A: set_real,X2: real] :
( ( finite_finite_real @ A )
=> ( ( ( member_real2 @ X2 @ A )
=> ( ( finite_card_real @ ( insert_real2 @ X2 @ A ) )
= ( finite_card_real @ A ) ) )
& ( ~ ( member_real2 @ X2 @ A )
=> ( ( finite_card_real @ ( insert_real2 @ X2 @ A ) )
= ( suc @ ( finite_card_real @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_834_card__insert__if,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) )
= ( finite711546835091564841at_nat @ A ) ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) )
= ( suc @ ( finite711546835091564841at_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_835_card__insert__if,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat2 @ X2 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_836_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: nat] :
( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_837_card__1__singleton__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ( finite711546835091564841at_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: product_prod_nat_nat] :
( A
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_838_card__1__singleton__iff,axiom,
! [A: set_real] :
( ( ( finite_card_real @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: real] :
( A
= ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).
% card_1_singleton_iff
thf(fact_839_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B3: nat,B6: set_nat] :
( ( A
= ( insert_nat2 @ B3 @ B6 ) )
& ~ ( member_nat2 @ B3 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_840_card__eq__SucD,axiom,
! [A: set_Pr1261947904930325089at_nat,K: nat] :
( ( ( finite711546835091564841at_nat @ A )
= ( suc @ K ) )
=> ? [B3: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ B3 @ B6 ) )
& ~ ( member8440522571783428010at_nat @ B3 @ B6 )
& ( ( finite711546835091564841at_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_841_card__eq__SucD,axiom,
! [A: set_real,K: nat] :
( ( ( finite_card_real @ A )
= ( suc @ K ) )
=> ? [B3: real,B6: set_real] :
( ( A
= ( insert_real2 @ B3 @ B6 ) )
& ~ ( member_real2 @ B3 @ B6 )
& ( ( finite_card_real @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_real ) ) ) ) ).
% card_eq_SucD
thf(fact_842_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B2: nat,B7: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ B7 ) )
& ~ ( member_nat2 @ B2 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_843_card__Suc__eq,axiom,
! [A: set_Pr1261947904930325089at_nat,K: nat] :
( ( ( finite711546835091564841at_nat @ A )
= ( suc @ K ) )
= ( ? [B2: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ B2 @ B7 ) )
& ~ ( member8440522571783428010at_nat @ B2 @ B7 )
& ( ( finite711546835091564841at_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_844_card__Suc__eq,axiom,
! [A: set_real,K: nat] :
( ( ( finite_card_real @ A )
= ( suc @ K ) )
= ( ? [B2: real,B7: set_real] :
( ( A
= ( insert_real2 @ B2 @ B7 ) )
& ~ ( member_real2 @ B2 @ B7 )
& ( ( finite_card_real @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_real ) ) ) ) ) ).
% card_Suc_eq
thf(fact_845_card__le__Suc__iff,axiom,
! [N: nat,A: set_real] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A ) )
= ( ? [A4: real,B7: set_real] :
( ( A
= ( insert_real2 @ A4 @ B7 ) )
& ~ ( member_real2 @ A4 @ B7 )
& ( ord_less_eq_nat @ N @ ( finite_card_real @ B7 ) )
& ( finite_finite_real @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_846_card__le__Suc__iff,axiom,
! [N: nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A ) )
= ( ? [A4: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
( ( A
= ( insert8211810215607154385at_nat @ A4 @ B7 ) )
& ~ ( member8440522571783428010at_nat @ A4 @ B7 )
& ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B7 ) )
& ( finite6177210948735845034at_nat @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_847_card__le__Suc__iff,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A ) )
= ( ? [A4: nat,B7: set_nat] :
( ( A
= ( insert_nat2 @ A4 @ B7 ) )
& ~ ( member_nat2 @ A4 @ B7 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B7 ) )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_848_set__replicate__conv__if,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_849_set__replicate__conv__if,axiom,
! [N: nat,X2: product_prod_nat_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X2 ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X2 ) )
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_850_set__replicate__conv__if,axiom,
! [N: nat,X2: real] :
( ( ( N = zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X2 ) )
= bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X2 ) )
= ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ) ).
% set_replicate_conv_if
thf(fact_851_set__replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_852_set__replicate__Suc,axiom,
! [N: nat,X2: product_prod_nat_nat] :
( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N ) @ X2 ) )
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).
% set_replicate_Suc
thf(fact_853_set__replicate__Suc,axiom,
! [N: nat,X2: real] :
( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X2 ) )
= ( insert_real2 @ X2 @ bot_bot_set_real ) ) ).
% set_replicate_Suc
thf(fact_854_sorted__sorted__list__of__multiset,axiom,
! [M4: multiset_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord3047872887403683810et_nat @ M4 ) ) ).
% sorted_sorted_list_of_multiset
thf(fact_855_sorted__sorted__list__of__multiset,axiom,
! [M4: multiset_real] : ( sorted_wrt_real @ ord_less_eq_real @ ( linord36121425647212990t_real @ M4 ) ) ).
% sorted_sorted_list_of_multiset
thf(fact_856_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_857_set__zero,axiom,
( zero_z7294763051868718104at_nat
= ( insert8211810215607154385at_nat @ zero_z3979849011205770936at_nat @ bot_bo2099793752762293965at_nat ) ) ).
% set_zero
thf(fact_858_set__zero,axiom,
( zero_zero_set_real
= ( insert_real2 @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_859_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_860_is__singletonI,axiom,
! [X2: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).
% is_singletonI
thf(fact_861_is__singletonI,axiom,
! [X2: real] : ( is_singleton_real @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ).
% is_singletonI
thf(fact_862_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( member_nat2 @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_863_is__singletonI_H,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( A != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A )
=> ( ( member8440522571783428010at_nat @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_sin2850979758926227957at_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_864_is__singletonI_H,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
=> ( ! [X3: real,Y: real] :
( ( member_real2 @ X3 @ A )
=> ( ( member_real2 @ Y @ A )
=> ( X3 = Y ) ) )
=> ( is_singleton_real @ A ) ) ) ).
% is_singletonI'
thf(fact_865_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_866_is__singletonE,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( is_sin2850979758926227957at_nat @ A )
=> ~ ! [X3: product_prod_nat_nat] :
( A
!= ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% is_singletonE
thf(fact_867_is__singletonE,axiom,
! [A: set_real] :
( ( is_singleton_real @ A )
=> ~ ! [X3: real] :
( A
!= ( insert_real2 @ X3 @ bot_bot_set_real ) ) ) ).
% is_singletonE
thf(fact_868_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_869_is__singleton__def,axiom,
( is_sin2850979758926227957at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat] :
? [X: product_prod_nat_nat] :
( A3
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% is_singleton_def
thf(fact_870_is__singleton__def,axiom,
( is_singleton_real
= ( ^ [A3: set_real] :
? [X: real] :
( A3
= ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).
% is_singleton_def
thf(fact_871_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_872_enumerate__Suc_H,axiom,
! [S: set_nat,N: nat] :
( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
= ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).
% enumerate_Suc'
thf(fact_873_card__Diff1__less__iff,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat2 @ X2 @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_874_card__Diff1__less__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) )
= ( ( finite6177210948735845034at_nat @ A )
& ( member8440522571783428010at_nat @ X2 @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_875_card__Diff1__less__iff,axiom,
! [A: set_real,X2: real] :
( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A ) )
= ( ( finite_finite_real @ A )
& ( member_real2 @ X2 @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_876_Diff__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B4 ) )
= ( ( member_nat2 @ C @ A )
& ~ ( member_nat2 @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_877_Diff__iff,axiom,
! [C: real,A: set_real,B4: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A @ B4 ) )
= ( ( member_real2 @ C @ A )
& ~ ( member_real2 @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_878_Diff__iff,axiom,
! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B4 ) )
= ( ( member8440522571783428010at_nat @ C @ A )
& ~ ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_879_DiffI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ~ ( member_nat2 @ C @ B4 )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_880_DiffI,axiom,
! [C: real,A: set_real,B4: set_real] :
( ( member_real2 @ C @ A )
=> ( ~ ( member_real2 @ C @ B4 )
=> ( member_real2 @ C @ ( minus_minus_set_real @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_881_DiffI,axiom,
! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ A )
=> ( ~ ( member8440522571783428010at_nat @ C @ B4 )
=> ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_882_semiring__norm_I58_J,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% semiring_norm(58)
thf(fact_883_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_884_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_885_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_886_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_887_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_888_right__minus__eq,axiom,
! [A2: real,B: real] :
( ( ( minus_minus_real @ A2 @ B )
= zero_zero_real )
= ( A2 = B ) ) ).
% right_minus_eq
thf(fact_889_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_890_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_891_Diff__empty,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A @ bot_bo2099793752762293965at_nat )
= A ) ).
% Diff_empty
thf(fact_892_Diff__empty,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ bot_bot_set_real )
= A ) ).
% Diff_empty
thf(fact_893_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_894_empty__Diff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A )
= bot_bo2099793752762293965at_nat ) ).
% empty_Diff
thf(fact_895_empty__Diff,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_896_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_897_Diff__cancel,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A @ A )
= bot_bo2099793752762293965at_nat ) ).
% Diff_cancel
thf(fact_898_Diff__cancel,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ A )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_899_finite__Diff2,axiom,
! [B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) )
= ( finite6177210948735845034at_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_900_finite__Diff2,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_901_finite__Diff,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_902_finite__Diff,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% finite_Diff
thf(fact_903_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ B4 ) )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_904_Diff__insert0,axiom,
! [X2: real,A: set_real,B4: set_real] :
( ~ ( member_real2 @ X2 @ A )
=> ( ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ B4 ) )
= ( minus_minus_set_real @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_905_Diff__insert0,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ B4 ) )
= ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_906_insert__Diff1,axiom,
! [X2: nat,B4: set_nat,A: set_nat] :
( ( member_nat2 @ X2 @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_907_insert__Diff1,axiom,
! [X2: real,B4: set_real,A: set_real] :
( ( member_real2 @ X2 @ B4 )
=> ( ( minus_minus_set_real @ ( insert_real2 @ X2 @ A ) @ B4 )
= ( minus_minus_set_real @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_908_insert__Diff1,axiom,
! [X2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) @ B4 )
= ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_909_diff__le__0__iff__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% diff_le_0_iff_le
thf(fact_910_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_911_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_912_diff__less__0__iff__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real )
= ( ord_less_real @ A2 @ B ) ) ).
% diff_less_0_iff_less
thf(fact_913_Diff__eq__empty__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_914_Diff__eq__empty__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A @ B4 )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_915_Diff__eq__empty__iff,axiom,
! [A: set_real,B4: set_real] :
( ( ( minus_minus_set_real @ A @ B4 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_916_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_917_insert__Diff__single,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
= ( insert8211810215607154385at_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_918_insert__Diff__single,axiom,
! [A2: real,A: set_real] :
( ( insert_real2 @ A2 @ ( minus_minus_set_real @ A @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) )
= ( insert_real2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_919_finite__Diff__insert,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B4 ) ) )
= ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_920_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_921_hd__in__set,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ( member_real2 @ ( hd_real @ Xs ) @ ( set_real2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_922_hd__in__set,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% hd_in_set
thf(fact_923_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_924_list_Oset__sel_I1_J,axiom,
! [A2: list_real] :
( ( A2 != nil_real )
=> ( member_real2 @ ( hd_real @ A2 ) @ ( set_real2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_925_list_Oset__sel_I1_J,axiom,
! [A2: list_P6011104703257516679at_nat] :
( ( A2 != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ A2 ) @ ( set_Pr5648618587558075414at_nat @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_926_list_Oset__sel_I1_J,axiom,
! [A2: list_nat] :
( ( A2 != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_927_diff__mono,axiom,
! [A2: real,B: real,D3: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D3 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_928_diff__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_929_diff__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_930_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C: real,D3: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_931_Diff__infinite__finite,axiom,
! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ T )
=> ( ~ ( finite6177210948735845034at_nat @ S )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_932_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_933_DiffD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( member_nat2 @ C @ B4 ) ) ).
% DiffD2
thf(fact_934_DiffD2,axiom,
! [C: real,A: set_real,B4: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A @ B4 ) )
=> ~ ( member_real2 @ C @ B4 ) ) ).
% DiffD2
thf(fact_935_DiffD2,axiom,
! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B4 ) )
=> ~ ( member8440522571783428010at_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_936_DiffD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ( member_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_937_DiffD1,axiom,
! [C: real,A: set_real,B4: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A @ B4 ) )
=> ( member_real2 @ C @ A ) ) ).
% DiffD1
thf(fact_938_DiffD1,axiom,
! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B4 ) )
=> ( member8440522571783428010at_nat @ C @ A ) ) ).
% DiffD1
thf(fact_939_DiffE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B4 ) ) ) ).
% DiffE
thf(fact_940_DiffE,axiom,
! [C: real,A: set_real,B4: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A @ B4 ) )
=> ~ ( ( member_real2 @ C @ A )
=> ( member_real2 @ C @ B4 ) ) ) ).
% DiffE
thf(fact_941_DiffE,axiom,
! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B4 ) )
=> ~ ( ( member8440522571783428010at_nat @ C @ A )
=> ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_942_psubset__imp__ex__mem,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ? [B3: nat] : ( member_nat2 @ B3 @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_943_psubset__imp__ex__mem,axiom,
! [A: set_real,B4: set_real] :
( ( ord_less_set_real @ A @ B4 )
=> ? [B3: real] : ( member_real2 @ B3 @ ( minus_minus_set_real @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_944_psubset__imp__ex__mem,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ B4 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_945_insert__Diff__if,axiom,
! [X2: nat,B4: set_nat,A: set_nat] :
( ( ( member_nat2 @ X2 @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) )
& ( ~ ( member_nat2 @ X2 @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B4 )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_946_insert__Diff__if,axiom,
! [X2: real,B4: set_real,A: set_real] :
( ( ( member_real2 @ X2 @ B4 )
=> ( ( minus_minus_set_real @ ( insert_real2 @ X2 @ A ) @ B4 )
= ( minus_minus_set_real @ A @ B4 ) ) )
& ( ~ ( member_real2 @ X2 @ B4 )
=> ( ( minus_minus_set_real @ ( insert_real2 @ X2 @ A ) @ B4 )
= ( insert_real2 @ X2 @ ( minus_minus_set_real @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_947_insert__Diff__if,axiom,
! [X2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) @ B4 )
= ( minus_1356011639430497352at_nat @ A @ B4 ) ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) @ B4 )
= ( insert8211810215607154385at_nat @ X2 @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_948_subset__Diff__insert,axiom,
! [A: set_nat,B4: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ ( insert_nat2 @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C2 ) )
& ~ ( member_nat2 @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_949_subset__Diff__insert,axiom,
! [A: set_real,B4: set_real,X2: real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B4 @ ( insert_real2 @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B4 @ C2 ) )
& ~ ( member_real2 @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_950_subset__Diff__insert,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( minus_1356011639430497352at_nat @ B4 @ ( insert8211810215607154385at_nat @ X2 @ C2 ) ) )
= ( ( ord_le3146513528884898305at_nat @ A @ ( minus_1356011639430497352at_nat @ B4 @ C2 ) )
& ~ ( member8440522571783428010at_nat @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_951_Diff__insert,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_952_Diff__insert,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B4 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% Diff_insert
thf(fact_953_Diff__insert,axiom,
! [A: set_real,A2: real,B4: set_real] :
( ( minus_minus_set_real @ A @ ( insert_real2 @ A2 @ B4 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A @ B4 ) @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_954_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_955_insert__Diff,axiom,
! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A2 @ A )
=> ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_956_insert__Diff,axiom,
! [A2: real,A: set_real] :
( ( member_real2 @ A2 @ A )
=> ( ( insert_real2 @ A2 @ ( minus_minus_set_real @ A @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) )
= A ) ) ).
% insert_Diff
thf(fact_957_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_958_Diff__insert2,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B4 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_959_Diff__insert2,axiom,
! [A: set_real,A2: real,B4: set_real] :
( ( minus_minus_set_real @ A @ ( insert_real2 @ A2 @ B4 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_960_Diff__insert__absorb,axiom,
! [X2: nat,A: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_961_Diff__insert__absorb,axiom,
! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_962_Diff__insert__absorb,axiom,
! [X2: real,A: set_real] :
( ~ ( member_real2 @ X2 @ A )
=> ( ( minus_minus_set_real @ ( insert_real2 @ X2 @ A ) @ ( insert_real2 @ X2 @ bot_bot_set_real ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_963_subset__insert__iff,axiom,
! [A: set_nat,X2: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B4 ) )
= ( ( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_964_subset__insert__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ B4 ) )
= ( ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_965_subset__insert__iff,axiom,
! [A: set_real,X2: real,B4: set_real] :
( ( ord_less_eq_set_real @ A @ ( insert_real2 @ X2 @ B4 ) )
= ( ( ( member_real2 @ X2 @ A )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) @ B4 ) )
& ( ~ ( member_real2 @ X2 @ A )
=> ( ord_less_eq_set_real @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_966_Diff__single__insert,axiom,
! [A: set_nat,X2: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B4 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_967_Diff__single__insert,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B4 )
=> ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_968_Diff__single__insert,axiom,
! [A: set_real,X2: real,B4: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) @ B4 )
=> ( ord_less_eq_set_real @ A @ ( insert_real2 @ X2 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_969_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat2 @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_970_finite__empty__induct,axiom,
! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A7 )
=> ( ( member8440522571783428010at_nat @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_971_finite__empty__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ A )
=> ( ! [A5: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( member_real2 @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ A5 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_972_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A7: set_nat] :
( ( X5 @ A7 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A7 )
& ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_973_infinite__coinduct,axiom,
! [X5: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
( ( X5 @ A )
=> ( ! [A7: set_Pr1261947904930325089at_nat] :
( ( X5 @ A7 )
=> ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A7 )
& ( ( X5 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) )
| ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ~ ( finite6177210948735845034at_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_974_infinite__coinduct,axiom,
! [X5: set_real > $o,A: set_real] :
( ( X5 @ A )
=> ( ! [A7: set_real] :
( ( X5 @ A7 )
=> ? [X4: real] :
( ( member_real2 @ X4 @ A7 )
& ( ( X5 @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X4 @ bot_bot_set_real ) ) )
| ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X4 @ bot_bot_set_real ) ) ) ) ) )
=> ~ ( finite_finite_real @ A ) ) ) ).
% infinite_coinduct
thf(fact_975_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_976_infinite__remove,axiom,
! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ~ ( finite6177210948735845034at_nat @ S )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% infinite_remove
thf(fact_977_infinite__remove,axiom,
! [S: set_real,A2: real] :
( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real2 @ A2 @ bot_bot_set_real ) ) ) ) ).
% infinite_remove
thf(fact_978_card__less__sym__Diff,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) )
=> ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_979_card__less__sym__Diff,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_980_card__le__sym__Diff,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( finite6177210948735845034at_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ B4 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B4 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_981_card__le__sym__Diff,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_982_remove__induct,axiom,
! [P: set_nat > $o,B4: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_983_remove__induct,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,B4: set_Pr1261947904930325089at_nat] :
( ( P @ bot_bo2099793752762293965at_nat )
=> ( ( ~ ( finite6177210948735845034at_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A7 )
=> ( ( A7 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A7 @ B4 )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A7 )
=> ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_984_remove__induct,axiom,
! [P: set_real > $o,B4: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B4 )
=> ( ! [X4: real] :
( ( member_real2 @ X4 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X4 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_985_finite__remove__induct,axiom,
! [B4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_986_finite__remove__induct,axiom,
! [B4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ B4 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A7 )
=> ( ( A7 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A7 @ B4 )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A7 )
=> ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_987_finite__remove__induct,axiom,
! [B4: set_real,P: set_real > $o] :
( ( finite_finite_real @ B4 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B4 )
=> ( ! [X4: real] :
( ( member_real2 @ X4 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X4 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_988_card__Diff1__le,axiom,
! [A: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% card_Diff1_le
thf(fact_989_card__Diff1__le,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) ) ).
% card_Diff1_le
thf(fact_990_card__Diff1__le,axiom,
! [A: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A ) ) ).
% card_Diff1_le
thf(fact_991_psubset__insert__iff,axiom,
! [A: set_nat,X2: nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat2 @ X2 @ B4 ) )
= ( ( ( member_nat2 @ X2 @ B4 )
=> ( ord_less_set_nat @ A @ B4 ) )
& ( ~ ( member_nat2 @ X2 @ B4 )
=> ( ( ( member_nat2 @ X2 @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_992_psubset__insert__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ B4 ) )
= ( ( ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ord_le7866589430770878221at_nat @ A @ B4 ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ A )
=> ( ord_le3146513528884898305at_nat @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_993_psubset__insert__iff,axiom,
! [A: set_real,X2: real,B4: set_real] :
( ( ord_less_set_real @ A @ ( insert_real2 @ X2 @ B4 ) )
= ( ( ( member_real2 @ X2 @ B4 )
=> ( ord_less_set_real @ A @ B4 ) )
& ( ~ ( member_real2 @ X2 @ B4 )
=> ( ( ( member_real2 @ X2 @ A )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) @ B4 ) )
& ( ~ ( member_real2 @ X2 @ A )
=> ( ord_less_eq_set_real @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_994_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T2: set_nat] :
( ( ord_less_set_nat @ T2 @ S )
=> ( ( P @ T2 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ ( minus_minus_set_nat @ S @ T2 ) )
& ( P @ ( insert_nat2 @ X4 @ T2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_995_finite__induct__select,axiom,
! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [T2: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ T2 @ S )
=> ( ( P @ T2 )
=> ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( minus_1356011639430497352at_nat @ S @ T2 ) )
& ( P @ ( insert8211810215607154385at_nat @ X4 @ T2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_996_finite__induct__select,axiom,
! [S: set_real,P: set_real > $o] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [T2: set_real] :
( ( ord_less_set_real @ T2 @ S )
=> ( ( P @ T2 )
=> ? [X4: real] :
( ( member_real2 @ X4 @ ( minus_minus_set_real @ S @ T2 ) )
& ( P @ ( insert_real2 @ X4 @ T2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_997_card__Suc__Diff1,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_998_card__Suc__Diff1,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
= ( finite711546835091564841at_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_999_card__Suc__Diff1,axiom,
! [A: set_real,X2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real2 @ X2 @ A )
=> ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) )
= ( finite_card_real @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1000_card_Oinsert__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X2 @ A ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1001_card_Oinsert__remove,axiom,
! [A: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A ) )
= ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1002_card_Oinsert__remove,axiom,
! [A: set_real,X2: real] :
( ( finite_finite_real @ A )
=> ( ( finite_card_real @ ( insert_real2 @ X2 @ A ) )
= ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1003_card_Oremove,axiom,
! [A: set_real,X2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real2 @ X2 @ A )
=> ( ( finite_card_real @ A )
= ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A @ ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1004_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1005_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1006_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1007_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1008_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1009_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1010_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1011_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1012_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1013_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1014_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1015_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1016_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1017_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1018_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1019_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1020_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1021_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1022_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1023_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1024_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1025_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1026_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1027_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1028_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1029_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1030_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1031_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1032_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1033_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1034_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1035_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1036_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1037_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_1038_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1039_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1040_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1041_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X4 @ I4 ) @ one_one_nat )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q @ I4 )
& ( ( X4 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X4 @ I4 )
= zero_zero_nat ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q @ I4 )
& ( ( X4 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X4 @ I4 )
= one_one_nat ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q @ I4 )
& ( ( L2 @ X4 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I4 ) @ ( F @ X4 @ I4 ) ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q @ I4 )
& ( ( L2 @ X4 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I4 ) @ ( X4 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1042_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real2 @ X4 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real2 @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y: real] :
( ! [X4: real] :
( ( member_real2 @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real2 @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1043_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1044_k__ge__1,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% k_ge_1
thf(fact_1045_seq__mono__lemma,axiom,
! [M2: nat,D3: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D3 @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M2 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ord_less_real @ ( D3 @ N5 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1046_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A5: real,B3: real,C4: real] :
( ( P @ A5 @ B3 )
=> ( ( P @ B3 @ C4 )
=> ( ( ord_less_eq_real @ A5 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C4 )
=> ( P @ A5 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A5: real,B3: real] :
( ( ( ord_less_eq_real @ A5 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A5 ) @ D ) )
=> ( P @ A5 @ B3 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1047_list__encode_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
% Conjectures (1)
thf(conj_0,conjecture,
( ( set_nat2 @ as )
!= bot_bot_set_nat ) ).
%------------------------------------------------------------------------------