TPTP Problem File: ITP229^1.p
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%------------------------------------------------------------------------------
% File : ITP229^1 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Insert 00617_038319
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0066_VEBT_Insert_00617_038319 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9649 (4246 unt; 920 typ; 0 def)
% Number of atoms : 26567 (10558 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 99778 (2694 ~; 466 |;1944 &;83472 @)
% ( 0 <=>;11202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 6 avg)
% Number of types : 82 ( 81 usr)
% Number of type conns : 4369 (4369 >; 0 *; 0 +; 0 <<)
% Number of symbols : 842 ( 839 usr; 70 con; 0-5 aty)
% Number of variables : 24039 (1945 ^;21344 !; 750 ?;24039 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 20:04:10.318
%------------------------------------------------------------------------------
% Could-be-implicit typings (81)
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_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__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
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% Explicit typings (839)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
archim2889992004027027881ng_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
archimedean_frac_rat: rat > rat ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_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__Num__Onum,type,
lattic1613168225601753569al_num: ( real > num ) > set_real > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
lattic4420706379359479199al_rat: ( real > rat ) > 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_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ocount__list_001_Eo,type,
count_list_o: list_o > $o > nat ).
thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
count_list_complex: list_complex > complex > nat ).
thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
count_list_int: list_int > int > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
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count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
count_list_real: list_real > real > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
count_list_set_nat: list_set_nat > set_nat > nat ).
thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).
thf(sy_c_List_Odistinct_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
fold_int_int: ( int > int > int ) > list_int > int > int ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
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set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
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set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
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set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
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size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__update_001_Eo,type,
list_update_o: list_o > nat > $o > list_o ).
thf(sy_c_List_Olist__update_001t__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
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list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
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list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
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list_update_real: list_real > nat > real > list_real ).
thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
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list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
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nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
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nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
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nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
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nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
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nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
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nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
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nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
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product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
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produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
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product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
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thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
remdups_nat: list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
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replicate_real: nat > real > list_real ).
thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
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replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
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thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
semiring_1_Nats_int: set_int ).
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semiri1314217659103216013at_int: nat > int ).
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semiri1316708129612266289at_nat: nat > nat ).
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semiri681578069525770553at_rat: nat > rat ).
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semiri5074537144036343181t_real: nat > real ).
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semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
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semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
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thf(sy_c_Nat__Bijection_Olist__encode,type,
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thf(sy_c_Nat__Bijection_Oset__encode,type,
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thf(sy_c_Nat__Bijection_Otriangle,type,
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thf(sy_c_NthRoot_Oroot,type,
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thf(sy_c_NthRoot_Osqrt,type,
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thf(sy_c_Num_Oinc,type,
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thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
dvd_dvd_rat: rat > rat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
zero_n3304061248610475627l_real: $o > real ).
thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
suminf_int: ( nat > int ) > int ).
thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
suminf_nat: ( nat > nat ) > nat ).
thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_001t__Int__Oint,type,
sums_int: ( nat > int ) > int > $o ).
thf(sy_c_Series_Osums_001t__Nat__Onat,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(sy_c_Series_Osums_001t__Real__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
insert_num: num > set_num > set_num ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
insert_rat: rat > set_rat > set_rat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or58775011639299419et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
cos_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
cosh_real: real > real ).
thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
cot_real: real > real ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
sin_real: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
sinh_real: real > real ).
thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
tan_real: real > real ).
thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
tanh_complex: complex > complex ).
thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
tanh_real: real > real ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
wf_int: set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: ( real > $o ) > real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $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__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,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_ya____,type,
ya: nat ).
% Relevant facts (8696)
thf(fact_0_False,axiom,
~ ( ord_less_nat @ mi @ ma ) ).
% False
thf(fact_1__092_060open_062x_A_060_Ami_092_060close_062,axiom,
ord_less_nat @ xa @ mi ).
% \<open>x < mi\<close>
thf(fact_2__C4_Ohyps_C_I7_J,axiom,
ord_less_eq_nat @ mi @ ma ).
% "4.hyps"(7)
thf(fact_3__C4_Oprems_C_I3_J,axiom,
vEBT_vebt_member @ ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ya ).
% "4.prems"(3)
thf(fact_4_True,axiom,
( ( ya = xa )
| ( ya
= ( ord_max_nat @ mi @ ma ) ) ) ).
% True
thf(fact_5__092_060open_062y_A_061_Amax_Ami_Ama_092_060close_062,axiom,
( ya
= ( ord_max_nat @ mi @ ma ) ) ).
% \<open>y = max mi ma\<close>
thf(fact_6_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_7_VEBT_Oinject_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_8_option_Oinject,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( some_P7363390416028606310at_nat @ X2 )
= ( some_P7363390416028606310at_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_9_option_Oinject,axiom,
! [X2: num,Y2: num] :
( ( ( some_num @ X2 )
= ( some_num @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
= ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ X1 @ X2 )
= ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
= ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ A @ B )
= ( produc9060074326276436823at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ A @ B )
= ( produc2922128104949294807at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_20__C4_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).
% "4.hyps"(6)
thf(fact_21_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K: nat,M: nat] :
( X
!= ( product_Pair_nat_nat @ K @ M ) ) ).
% prod_decode_aux.cases
thf(fact_22__C4_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "4.hyps"(1)
thf(fact_23_old_Oprod_Oexhaust,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( Y
!= ( produc6161850002892822231at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
! [Y: produc1319942482725812455at_nat] :
~ ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
( Y
!= ( produc9060074326276436823at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_25_old_Oprod_Oexhaust,axiom,
! [Y: produc3843707927480180839at_nat] :
~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( Y
!= ( produc2922128104949294807at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_26_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_27_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_28_surj__pair,axiom,
! [P: produc859450856879609959at_nat] :
? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( P
= ( produc6161850002892822231at_nat @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_29_surj__pair,axiom,
! [P: produc1319942482725812455at_nat] :
? [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( P
= ( produc9060074326276436823at_nat @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_30_surj__pair,axiom,
! [P: produc3843707927480180839at_nat] :
? [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( P
= ( produc2922128104949294807at_nat @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_31_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X4: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_32_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X4: int,Y3: int] :
( P
= ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_33_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_34_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).
% not_min_Null_member
thf(fact_35_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_vebt_member @ T @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_36_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
= ( vEBT_vebt_member @ T @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_37_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ T @ X )
= ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% member_correct
thf(fact_38_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_39_prod__induct3,axiom,
! [P2: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_40_prod__cases3,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] :
( Y
!= ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_41_Pair__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_42_Pair__inject,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ A @ B )
= ( produc9060074326276436823at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_43_Pair__inject,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ A @ B )
= ( produc2922128104949294807at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_44_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_46_prod__cases,axiom,
! [P2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat] :
( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_47_prod__cases,axiom,
! [P2: produc1319942482725812455at_nat > $o,P: produc1319942482725812455at_nat] :
( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( P2 @ ( produc9060074326276436823at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_48_prod__cases,axiom,
! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_49_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_50_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_51__C4_Ohyps_C_I3_J,axiom,
m = na ).
% "4.hyps"(3)
thf(fact_52_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_53_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_54_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_55_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_56_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_57_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_58_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_59_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_60_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_61_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_62_max__less__iff__conj,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_63_max__less__iff__conj,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ( ord_less_rat @ X @ Z )
& ( ord_less_rat @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_64_max__less__iff__conj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
& ( ord_less_num @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_65_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
& ( ord_less_nat @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_66_max__less__iff__conj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
& ( ord_less_int @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_67_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_68_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_69_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_70_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_71_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_72_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_73_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_74_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_75_max_Obounded__iff,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
= ( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_76_max_Obounded__iff,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_77_max_Obounded__iff,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_78_max_Obounded__iff,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_79_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_80_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_set_vebt @ T )
= ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_81_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X5: real] : ( member_real @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A4: set_set_nat] :
( ( collect_set_nat
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X5: int] : ( member_int @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_93_Collect__cong,axiom,
! [P2: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_real @ P2 )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_94_Collect__cong,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_nat @ P2 )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_95_Collect__cong,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P2 )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_96_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_97_Collect__cong,axiom,
! [P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P2 )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_98_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_99_max_Oidem,axiom,
! [A: int] :
( ( ord_max_int @ A @ A )
= A ) ).
% max.idem
thf(fact_100_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_101_max_Oleft__idem,axiom,
! [A: int,B: int] :
( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
= ( ord_max_int @ A @ B ) ) ).
% max.left_idem
thf(fact_102_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_103_max_Oright__idem,axiom,
! [A: int,B: int] :
( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
= ( ord_max_int @ A @ B ) ) ).
% max.right_idem
thf(fact_104__C4_Ohyps_C_I4_J,axiom,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
% "4.hyps"(4)
thf(fact_105_max_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C2 ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C2 ) ) ) ).
% max.left_commute
thf(fact_106_max_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( ord_max_int @ B @ ( ord_max_int @ A @ C2 ) )
= ( ord_max_int @ A @ ( ord_max_int @ B @ C2 ) ) ) ).
% max.left_commute
thf(fact_107_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( ord_max_nat @ B4 @ A5 ) ) ) ).
% max.commute
thf(fact_108_max_Ocommute,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( ord_max_int @ B4 @ A5 ) ) ) ).
% max.commute
thf(fact_109_max_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C2 )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C2 ) ) ) ).
% max.assoc
thf(fact_110_max_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C2 )
= ( ord_max_int @ A @ ( ord_max_int @ B @ C2 ) ) ) ).
% max.assoc
thf(fact_111_max_OcoboundedI2,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_112_max_OcoboundedI2,axiom,
! [C2: num,B: num,A: num] :
( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_113_max_OcoboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_114_max_OcoboundedI2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_115_max_OcoboundedI1,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ A )
=> ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_116_max_OcoboundedI1,axiom,
! [C2: num,A: num,B: num] :
( ( ord_less_eq_num @ C2 @ A )
=> ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_117_max_OcoboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_118_max_OcoboundedI1,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_119_max_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_max_rat @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_120_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_max_num @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_121_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_max_nat @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_122_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_max_int @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_123_max_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_max_rat @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_124_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_max_num @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_125_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_max_nat @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_126_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_max_int @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_127_le__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
= ( ( ord_less_eq_rat @ Z @ X )
| ( ord_less_eq_rat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_128_le__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_eq_num @ Z @ X )
| ( ord_less_eq_num @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_129_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z @ X )
| ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_130_le__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_eq_int @ Z @ X )
| ( ord_less_eq_int @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_131_max_Ocobounded2,axiom,
! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded2
thf(fact_132_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_133_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_134_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_135_max_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded1
thf(fact_136_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_137_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_138_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_139_max_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( A5
= ( ord_max_rat @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_140_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( A5
= ( ord_max_num @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_141_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( ord_max_nat @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_142_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( A5
= ( ord_max_int @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_143_max_OboundedI,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ A )
=> ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_144_max_OboundedI,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_145_max_OboundedI,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_146_max_OboundedI,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_147_max_OboundedE,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_rat @ B @ A )
=> ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_148_max_OboundedE,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_149_max_OboundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_150_max_OboundedE,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_151_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( ord_max_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% max.orderI
thf(fact_152_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_153_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_154_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_155_max_OorderE,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( A
= ( ord_max_rat @ A @ B ) ) ) ).
% max.orderE
thf(fact_156_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_157_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_158_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_159_max_Omono,axiom,
! [C2: rat,A: rat,D: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ A )
=> ( ( ord_less_eq_rat @ D @ B )
=> ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_160_max_Omono,axiom,
! [C2: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C2 @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C2 @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_161_max_Omono,axiom,
! [C2: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_162_max_Omono,axiom,
! [C2: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C2 @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_163_max_Ostrict__coboundedI2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_164_max_Ostrict__coboundedI2,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_165_max_Ostrict__coboundedI2,axiom,
! [C2: num,B: num,A: num] :
( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_166_max_Ostrict__coboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_167_max_Ostrict__coboundedI2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_168_max_Ostrict__coboundedI1,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ A )
=> ( ord_less_real @ C2 @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_169_max_Ostrict__coboundedI1,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ A )
=> ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_170_max_Ostrict__coboundedI1,axiom,
! [C2: num,A: num,B: num] :
( ( ord_less_num @ C2 @ A )
=> ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_171_max_Ostrict__coboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ C2 @ A )
=> ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_172_max_Ostrict__coboundedI1,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ A )
=> ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_173_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( A5
= ( ord_max_real @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_174_max_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( A5
= ( ord_max_rat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_175_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( A5
= ( ord_max_num @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_176_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( A5
= ( ord_max_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_177_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( A5
= ( ord_max_int @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_178_max_Ostrict__boundedE,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( ord_max_real @ B @ C2 ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C2 @ A ) ) ) ).
% max.strict_boundedE
thf(fact_179_max_Ostrict__boundedE,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_rat @ ( ord_max_rat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ C2 @ A ) ) ) ).
% max.strict_boundedE
thf(fact_180_max_Ostrict__boundedE,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_num @ ( ord_max_num @ B @ C2 ) @ A )
=> ~ ( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ C2 @ A ) ) ) ).
% max.strict_boundedE
thf(fact_181_max_Ostrict__boundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C2 @ A ) ) ) ).
% max.strict_boundedE
thf(fact_182_max_Ostrict__boundedE,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_int @ ( ord_max_int @ B @ C2 ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C2 @ A ) ) ) ).
% max.strict_boundedE
thf(fact_183_less__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_real @ Z @ X )
| ( ord_less_real @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_184_less__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
= ( ( ord_less_rat @ Z @ X )
| ( ord_less_rat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_185_less__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
| ( ord_less_num @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_186_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
| ( ord_less_nat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_187_less__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
| ( ord_less_int @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_188_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_189_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_190_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_191_dual__order_Orefl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% dual_order.refl
thf(fact_192_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_193_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_194_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_195_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_196_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_197_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_198_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_199_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_200_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_201_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_finite
thf(fact_202_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% deg_not_0
thf(fact_203_max__absorb2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_max_set_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_204_max__absorb2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_max_rat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_205_max__absorb2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_max_num @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_206_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_207_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_208_max__absorb1,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_max_set_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_209_max__absorb1,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_max_rat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_210_max__absorb1,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_max_num @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_211_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_212_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_213_max__def,axiom,
( ord_max_set_int
= ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_214_max__def,axiom,
( ord_max_rat
= ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_215_max__def,axiom,
( ord_max_num
= ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_216_max__def,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_217_max__def,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_218_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K @ I )
=> ( P2 @ I ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_219_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_220_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_221_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_222_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_223_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_224_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_225_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_226_le__cases3,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_227_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_228_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_229_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_230_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ( ord_less_eq_set_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_231_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ( ord_less_eq_rat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_232_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_233_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_234_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_235_ord__eq__le__trans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( A = B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_237_ord__eq__le__trans,axiom,
! [A: num,B: num,C2: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_238_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_239_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_240_ord__le__eq__trans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_241_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_242_ord__le__eq__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_243_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_244_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_245_order__antisym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_246_order__antisym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_247_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_248_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_249_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_250_order_Otrans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_251_order_Otrans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_252_order_Otrans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% order.trans
thf(fact_253_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_254_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_255_order__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_256_order__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_259_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_260_linorder__wlog,axiom,
! [P2: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: rat,B3: rat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_261_linorder__wlog,axiom,
! [P2: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: num,B3: num] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_262_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_263_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_264_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_265_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_266_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_267_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_268_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_269_dual__order_Oantisym,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_270_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_271_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_272_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_273_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_274_dual__order_Otrans,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C2 @ B )
=> ( ord_less_eq_set_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_275_dual__order_Otrans,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_276_dual__order_Otrans,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_eq_num @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_277_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_278_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_279_antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_280_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_281_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_282_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_283_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_284_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_285_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_286_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_287_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_288_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_289_order__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_290_order__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_291_order__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_292_order__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_293_order__subst1,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_294_order__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_295_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_296_order__subst1,axiom,
! [A: num,F: int > num,B: int,C2: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_297_order__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_298_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_299_order__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_300_order__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_301_order__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_302_order__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_303_order__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_304_order__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_305_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_306_order__subst2,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_307_order__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_308_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_309_order__eq__refl,axiom,
! [X: set_int,Y: set_int] :
( ( X = Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_310_order__eq__refl,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_311_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_312_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_313_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_314_linorder__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_linear
thf(fact_315_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_316_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_317_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_318_ord__eq__le__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_319_ord__eq__le__subst,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_320_ord__eq__le__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_321_ord__eq__le__subst,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_322_ord__eq__le__subst,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_323_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_324_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_325_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_326_ord__eq__le__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_327_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_328_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_329_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_330_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_331_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_332_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_333_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_334_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_335_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_336_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_337_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_338_linorder__le__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_339_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_340_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_341_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_342_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_343_order__antisym__conv,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_344_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_345_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_346_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_347_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_348_lt__ex,axiom,
! [X: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% lt_ex
thf(fact_349_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_350_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_351_gt__ex,axiom,
! [X: rat] :
? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).
% gt_ex
thf(fact_352_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_353_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_354_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_355_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_356_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_357_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_358_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_359_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_360_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_361_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_362_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_363_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_364_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_365_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_366_ord__eq__less__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_367_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_368_ord__eq__less__trans,axiom,
! [A: num,B: num,C2: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_369_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_370_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_371_ord__less__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_372_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_373_ord__less__eq__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_374_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_375_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_376_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_377_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_378_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_379_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_380_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_381_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_382_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_383_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_384_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_385_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_386_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_387_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_388_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_389_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_390_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_391_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_392_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_393_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_394_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_395_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_396_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_397_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_398_linorder__less__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P2 @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_399_linorder__less__wlog,axiom,
! [P2: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: rat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: rat,B3: rat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_400_linorder__less__wlog,axiom,
! [P2: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: num] : ( P2 @ A3 @ A3 )
=> ( ! [A3: num,B3: num] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_401_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_402_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P2 @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_403_order_Ostrict__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_404_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_405_order_Ostrict__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_406_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_407_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_408_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_409_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_410_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_411_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_412_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_413_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_414_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_415_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_416_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_417_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_418_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_419_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_420_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_421_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_422_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_423_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_424_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_425_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_426_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_427_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_428_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_429_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_430_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_431_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_432_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_433_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_434_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_435_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_436_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_437_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_438_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_439_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_440_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_441_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_442_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_443_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_444_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_445_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_446_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_447_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_448_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_449_order__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_450_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_451_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_452_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_453_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_454_ord__eq__less__subst,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_455_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_456_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_457_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_458_ord__eq__less__subst,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_459_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_460_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_461_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_462_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_463_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_464_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_465_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C2: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_466_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_467_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_468_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_469_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_470_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_471_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_472_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_473_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_474_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_475_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_476_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_477_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_478_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_479_order__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_480_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_481_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_482_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_483_order__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_484_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_485_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_486_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_487_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_488_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_489_order__less__subst2,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_490_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C2: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_491_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_492_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_493_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_494_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_495_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_496_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_497_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_498_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_499_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_500_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_501_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_502_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_503_order__less__imp__triv,axiom,
! [X: real,Y: real,P2: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_504_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P2: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_505_order__less__imp__triv,axiom,
! [X: num,Y: num,P2: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_506_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_507_order__less__imp__triv,axiom,
! [X: int,Y: int,P2: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_508_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_509_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_510_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_511_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_512_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_513_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_514_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_515_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_516_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_517_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_518_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_519_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_520_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_521_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_522_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_523_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_524_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_525_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_526_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_527_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_528_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_529_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_530_leD,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ~ ( ord_less_set_int @ X @ Y ) ) ).
% leD
thf(fact_531_leD,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_rat @ X @ Y ) ) ).
% leD
thf(fact_532_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_533_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_534_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_535_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_536_leI,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% leI
thf(fact_537_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_538_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_539_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_540_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_541_nless__le,axiom,
! [A: set_int,B: set_int] :
( ( ~ ( ord_less_set_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_542_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_543_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_544_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_545_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_546_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_547_antisym__conv1,axiom,
! [X: set_int,Y: set_int] :
( ~ ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_548_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_549_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_550_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_551_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_552_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_553_antisym__conv2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ~ ( ord_less_set_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_554_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_555_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_556_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_557_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_558_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_559_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ Z @ X4 )
=> ( ord_less_eq_rat @ Y @ X4 ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_ge
thf(fact_560_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_561_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ X4 @ Y )
=> ( ord_less_eq_rat @ X4 @ Z ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_le
thf(fact_562_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_563_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ~ ( ord_less_eq_set_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_564_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ~ ( ord_less_eq_rat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_565_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_566_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_567_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_568_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_569_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_rat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_570_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_571_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_572_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_573_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_574_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_set_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_575_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_576_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_577_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_578_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_579_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_580_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_581_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_582_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_583_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_584_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_585_order_Ostrict__trans1,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_586_order_Ostrict__trans1,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_set_int @ B @ C2 )
=> ( ord_less_set_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_587_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_588_order_Ostrict__trans1,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_589_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_590_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_591_order_Ostrict__trans2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_592_order_Ostrict__trans2,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_set_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_593_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_594_order_Ostrict__trans2,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_595_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_596_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_597_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_598_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_599_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_600_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_601_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_602_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_603_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_604_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z @ W )
=> ( ( ord_less_rat @ W @ X )
=> ( ord_less_eq_rat @ Y @ W ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_605_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_606_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ! [W: rat] :
( ( ord_less_rat @ X @ W )
=> ( ( ord_less_rat @ W @ Y )
=> ( ord_less_eq_rat @ W @ Z ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_607_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_608_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_set_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_609_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_610_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_611_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_612_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_613_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_614_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_615_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_616_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_617_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_618_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_619_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_620_dual__order_Ostrict__trans1,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_621_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_622_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_623_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_624_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_625_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_626_dual__order_Ostrict__trans2,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_627_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_628_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_629_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_630_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_631_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_632_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_633_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_634_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_635_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_636_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_637_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_638_order_Ostrict__implies__order,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_639_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_640_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_641_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_642_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_643_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_644_dual__order_Ostrict__implies__order,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ord_less_eq_set_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_645_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_646_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_647_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_648_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_649_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_650_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_set_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_651_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_rat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_652_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_653_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_654_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_655_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_656_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_657_order__less__le,axiom,
( ord_less_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_658_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_659_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_660_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_661_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_662_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y ) )
= ( ord_less_rat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_663_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_664_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_665_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_666_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_667_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_668_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_669_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_670_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_671_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_672_order__less__imp__le,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_673_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_674_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_675_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_676_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_677_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_678_order__le__neq__trans,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_679_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_680_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_681_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_682_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_683_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_684_order__neq__le__trans,axiom,
! [A: set_int,B: set_int] :
( ( A != B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_685_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_686_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_687_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_688_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_689_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_690_order__le__less__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_691_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_692_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_693_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_694_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_695_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_696_order__less__le__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_697_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_698_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_699_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_700_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_701_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_702_order__le__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_703_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_704_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_705_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_706_order__le__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_707_order__le__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_708_order__le__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_709_order__le__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_710_order__le__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_711_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_712_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_713_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_714_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_715_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_716_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C2: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_717_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_718_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_719_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_720_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_721_order__less__le__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_722_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_723_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_724_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_725_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_726_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_727_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_728_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_729_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_730_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C2: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_731_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_732_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_733_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C2: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_734_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_735_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_736_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_737_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_738_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_739_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_740_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C2: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_741_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_742_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_743_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_744_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_745_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_746_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_747_order__le__imp__less__or__eq,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_748_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_749_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_750_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_751_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_752_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_753_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_754_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_755_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_756_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_757_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_758_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_759_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_760_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_761_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_762_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_763_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_764_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_765_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_766_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_767_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_768_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_769_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_770_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_771_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_772_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_773_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_774_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_775_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_776_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_777_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_778_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_779_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_780_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_781_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_782_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_783_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_784_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_785_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_786_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_787_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_788_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_789_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_790_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_791_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_792_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_793_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_794_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_795_add__right__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_796_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_797_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_798_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_799_add__left__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_800_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_801_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_802_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_803_add__le__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_804_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_805_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_806_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_807_add__le__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_808_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_809_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_810_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_811_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_812_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_813_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_814_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_815_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_816_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_817_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_818_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_819_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_820_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_821_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_822_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_823_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_824_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_825_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_826_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_827_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_828_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_829_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_830_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_831_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_832_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_833_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_834_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_835_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_836_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_837_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_838_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_839_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_840_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_841_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_842_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_843_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_844_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_845_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_846_add__less__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_847_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_848_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_849_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_850_add__less__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_851_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_852_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_853_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_854_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_855_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_856_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_857_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_858_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_859_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_860_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_861_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_862_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_863_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_864_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_865_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_866_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_867_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_868_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_869_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_870_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_871_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_872_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_873_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_874_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_875_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_876_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_877_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_878_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_879_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_880_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_881_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_882_add__right__imp__eq,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_883_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_884_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_885_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_886_add__left__imp__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_887_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_888_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_889_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_890_add_Oleft__commute,axiom,
! [B: rat,A: rat,C2: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_891_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_892_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_893_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_894_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_895_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_896_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_897_add_Oright__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_898_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_899_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_900_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_901_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_902_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_903_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_904_add_Oassoc,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_905_group__cancel_Oadd2,axiom,
! [B5: nat,K2: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_906_group__cancel_Oadd2,axiom,
! [B5: int,K2: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_907_group__cancel_Oadd2,axiom,
! [B5: real,K2: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K2 @ B ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_908_group__cancel_Oadd2,axiom,
! [B5: rat,K2: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K2 @ B ) )
=> ( ( plus_plus_rat @ A @ B5 )
= ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_909_group__cancel_Oadd1,axiom,
! [A4: nat,K2: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_910_group__cancel_Oadd1,axiom,
! [A4: int,K2: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_911_group__cancel_Oadd1,axiom,
! [A4: real,K2: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_912_group__cancel_Oadd1,axiom,
! [A4: rat,K2: rat,A: rat,B: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( plus_plus_rat @ A4 @ B )
= ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_913_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I2 @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_914_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I2 @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_915_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_real @ I2 @ K2 )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_916_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_rat @ I2 @ K2 )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_917_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_918_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_919_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_920_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_921_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_922_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_923_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_924_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_925_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_926_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_927_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_928_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_929_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_930_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_931_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_932_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_933_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_934_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_935_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_936_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_937_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_938_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_939_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_940_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_941_add__le__imp__le__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_942_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_943_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_944_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_945_add__le__imp__le__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_946_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_947_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_948_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_949_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_950_add__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_951_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_952_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_953_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_954_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_955_add__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_956_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_957_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_958_add__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_959_add__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_960_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_961_add__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_962_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_963_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_964_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_965_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_966_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_967_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( I2 = J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_968_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_969_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_970_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_971_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_972_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_973_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_974_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_975_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_976_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_977_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_978_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_979_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_980_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_981_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_982_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_983_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_984_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_985_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_986_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_987_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_988_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_989_add__less__imp__less__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_990_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_991_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_992_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_993_add__less__imp__less__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_994_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_995_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_996_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_997_add__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_998_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_999_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1000_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1001_add__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1002_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1003_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1004_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1005_add__strict__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1006_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1007_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1008_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1009_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1010_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1011_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1012_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1013_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( I2 = J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1014_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1015_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1016_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1017_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1018_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1019_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1020_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1021_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1022_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1023_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1024_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1025_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1026_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1027_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1028_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1029_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1030_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1031_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_1032_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_1033_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_1034_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1035_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1036_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1037_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1038_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1039_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1040_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1041_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1042_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1043_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1044_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1045_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1046_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
=> ( ord_less_nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_1047_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1048_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1049_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1050_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1051_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1052_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1053_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1054_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1055_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1056_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1057_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1058_max__add__distrib__right,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
= ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1059_max__add__distrib__right,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1060_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1061_max__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1062_max__add__distrib__left,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1063_max__add__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1064_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1065_max__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1066_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1067_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1068_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1069_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1070_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1071_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1072_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1073_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1074_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1075_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1076_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1077_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1078_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1079_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1080_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1081_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1082_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1083_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1084_add__increasing2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1085_add__increasing2,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1086_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1087_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1088_add__decreasing2,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1089_add__decreasing2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1090_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1091_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1092_add__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1093_add__increasing,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1094_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1095_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1096_add__decreasing,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1097_add__decreasing,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1098_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1099_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1100_add__less__le__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1101_add__less__le__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1102_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1103_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1104_add__le__less__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1105_add__le__less__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1106_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1107_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1108_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1109_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1110_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1111_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1112_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1113_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1114_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1115_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1116_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1117_pos__add__strict,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1118_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1119_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1120_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1121_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1122_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1123_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1124_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1125_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1126_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_1127_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1128_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1129_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1130_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I2 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1131_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M: nat,N3: nat] :
( ( ord_less_nat @ M @ N3 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1132_add__strict__increasing2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1133_add__strict__increasing2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1134_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1135_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1136_add__strict__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1137_add__strict__increasing,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1138_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1139_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1140_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1141_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1142_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1143_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1144_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1145_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_1146_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1147_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1148_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1149_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1150_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1151_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1152_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1153_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_1154_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1155_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1156_List_Ofinite__set,axiom,
! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% List.finite_set
thf(fact_1157_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1158_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_1159_List_Ofinite__set,axiom,
! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% List.finite_set
thf(fact_1160_List_Ofinite__set,axiom,
! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_1161_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_1162_double__eq__0__iff,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_1163_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1164_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_1165_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1166_field__le__epsilon,axiom,
! [X: rat,Y: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1167_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1168_add__less__zeroD,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_1169_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1170_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_1171_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_1172_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1173_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1174_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N4 )
=> ( ord_less_nat @ X5 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1175_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T2: vEBT_VEBT,X5: nat] :
( ( vEBT_V5719532721284313246member @ T2 @ X5 )
| ( vEBT_VEBT_membermima @ T2 @ X5 ) ) ) ) ).
% both_member_options_def
thf(fact_1176_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1177_max__bot,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X )
= X ) ).
% max_bot
thf(fact_1178_max__bot,axiom,
! [X: set_real] :
( ( ord_max_set_real @ bot_bot_set_real @ X )
= X ) ).
% max_bot
thf(fact_1179_max__bot,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% max_bot
thf(fact_1180_max__bot,axiom,
! [X: set_int] :
( ( ord_max_set_int @ bot_bot_set_int @ X )
= X ) ).
% max_bot
thf(fact_1181_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_1182_max__bot2,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( ord_ma7524802468073614006at_nat @ X @ bot_bo2099793752762293965at_nat )
= X ) ).
% max_bot2
thf(fact_1183_max__bot2,axiom,
! [X: set_real] :
( ( ord_max_set_real @ X @ bot_bot_set_real )
= X ) ).
% max_bot2
thf(fact_1184_max__bot2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% max_bot2
thf(fact_1185_max__bot2,axiom,
! [X: set_int] :
( ( ord_max_set_int @ X @ bot_bot_set_int )
= X ) ).
% max_bot2
thf(fact_1186_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_1187_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_1188_finite_OemptyI,axiom,
finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% finite.emptyI
thf(fact_1189_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_1190_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1191_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_1192_infinite__imp__nonempty,axiom,
! [S2: set_complex] :
( ~ ( finite3207457112153483333omplex @ S2 )
=> ( S2 != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_1193_infinite__imp__nonempty,axiom,
! [S2: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S2 )
=> ( S2 != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1194_infinite__imp__nonempty,axiom,
! [S2: set_real] :
( ~ ( finite_finite_real @ S2 )
=> ( S2 != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_1195_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1196_infinite__imp__nonempty,axiom,
! [S2: set_int] :
( ~ ( finite_finite_int @ S2 )
=> ( S2 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_1197_finite__transitivity__chain,axiom,
! [A4: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [X4: set_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: set_nat,Y3: set_nat,Z3: set_nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1198_finite__transitivity__chain,axiom,
! [A4: set_complex,R: complex > complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: complex,Y3: complex,Z3: complex] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Y6: complex] :
( ( member_complex @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_complex ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1199_finite__transitivity__chain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [X4: product_prod_nat_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ? [Y6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1200_finite__transitivity__chain,axiom,
! [A4: set_real,R: real > real > $o] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: real,Y3: real,Z3: real] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Y6: real] :
( ( member_real @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1201_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1202_finite__transitivity__chain,axiom,
! [A4: set_int,R: int > int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: int,Y3: int,Z3: int] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Y6: int] :
( ( member_int @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1203_bot_Oextremum,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% bot.extremum
thf(fact_1204_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_1205_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1206_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_1207_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1208_bot_Oextremum__unique,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_unique
thf(fact_1209_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_1210_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1211_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_1212_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1213_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1214_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_1215_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1216_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_1217_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1218_bot_Oextremum__strict,axiom,
! [A: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% bot.extremum_strict
thf(fact_1219_bot_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_1220_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1221_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_1222_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1223_bot_Onot__eq__extremum,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( A != bot_bo2099793752762293965at_nat )
= ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1224_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1225_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1226_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1227_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1228_finite__has__maximal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1229_finite__has__maximal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1230_finite__has__maximal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1231_finite__has__maximal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1232_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1233_finite__has__maximal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1234_finite__has__minimal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1235_finite__has__minimal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1236_finite__has__minimal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1237_finite__has__minimal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1238_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1239_finite__has__minimal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1240_infinite__growing,axiom,
! [X7: set_real] :
( ( X7 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X7 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X7 )
& ( ord_less_real @ X4 @ Xa ) ) )
=> ~ ( finite_finite_real @ X7 ) ) ) ).
% infinite_growing
thf(fact_1241_infinite__growing,axiom,
! [X7: set_rat] :
( ( X7 != bot_bot_set_rat )
=> ( ! [X4: rat] :
( ( member_rat @ X4 @ X7 )
=> ? [Xa: rat] :
( ( member_rat @ Xa @ X7 )
& ( ord_less_rat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_rat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1242_infinite__growing,axiom,
! [X7: set_num] :
( ( X7 != bot_bot_set_num )
=> ( ! [X4: num] :
( ( member_num @ X4 @ X7 )
=> ? [Xa: num] :
( ( member_num @ Xa @ X7 )
& ( ord_less_num @ X4 @ Xa ) ) )
=> ~ ( finite_finite_num @ X7 ) ) ) ).
% infinite_growing
thf(fact_1243_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1244_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X7 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X7 )
& ( ord_less_int @ X4 @ Xa ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_1245_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1246_ex__min__if__finite,axiom,
! [S2: set_rat] :
( ( finite_finite_rat @ S2 )
=> ( ( S2 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ S2 )
& ~ ? [Xa: rat] :
( ( member_rat @ Xa @ S2 )
& ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1247_ex__min__if__finite,axiom,
! [S2: set_num] :
( ( finite_finite_num @ S2 )
=> ( ( S2 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ S2 )
& ~ ? [Xa: num] :
( ( member_num @ Xa @ S2 )
& ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1248_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1249_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1250_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1251_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).
% linordered_field_no_lb
thf(fact_1252_linordered__field__no__lb,axiom,
! [X3: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X3 ) ).
% linordered_field_no_lb
thf(fact_1253_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_1254_linordered__field__no__ub,axiom,
! [X3: rat] :
? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_1255_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1256_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1257_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1258_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_1259_finite__subset,axiom,
! [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_subset
thf(fact_1260_finite__subset,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_1261_finite__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_subset
thf(fact_1262_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1263_infinite__super,axiom,
! [S2: set_complex,T3: set_complex] :
( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% infinite_super
thf(fact_1264_infinite__super,axiom,
! [S2: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ S2 @ T3 )
=> ( ~ ( finite6177210948735845034at_nat @ S2 )
=> ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1265_infinite__super,axiom,
! [S2: set_int,T3: set_int] :
( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ T3 ) ) ) ).
% infinite_super
thf(fact_1266_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1267_rev__finite__subset,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1268_rev__finite__subset,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1269_rev__finite__subset,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1270_finite__psubset__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1271_finite__psubset__induct,axiom,
! [A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [B6: set_int] :
( ( ord_less_set_int @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1272_finite__psubset__induct,axiom,
! [A4: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [A6: set_complex] :
( ( finite3207457112153483333omplex @ A6 )
=> ( ! [B6: set_complex] :
( ( ord_less_set_complex @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1273_finite__psubset__induct,axiom,
! [A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [A6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A6 )
=> ( ! [B6: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1274_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M5: nat] :
( ( P2 @ X )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M: nat] :
( ( P2 @ M )
=> ~ ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1275_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B5 )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1276_subset__code_I1_J,axiom,
! [Xs: list_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B5 )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
=> ( member_real @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1277_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B5 )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1278_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B5 )
= ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( member_VEBT_VEBT @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1279_subset__code_I1_J,axiom,
! [Xs: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1280_subset__code_I1_J,axiom,
! [Xs: list_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B5 )
= ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
=> ( member_int @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1281_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1282_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1283_finite__has__maximal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ A @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1284_finite__has__maximal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ A @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1285_finite__has__maximal2,axiom,
! [A4: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ A @ X4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1286_finite__has__maximal2,axiom,
! [A4: set_rat,A: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ A @ X4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1287_finite__has__maximal2,axiom,
! [A4: set_num,A: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ A @ X4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1288_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1289_finite__has__maximal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ A @ X4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1290_finite__has__minimal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ X4 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1291_finite__has__minimal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ X4 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1292_finite__has__minimal2,axiom,
! [A4: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ X4 @ A )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1293_finite__has__minimal2,axiom,
! [A4: set_rat,A: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ X4 @ A )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1294_finite__has__minimal2,axiom,
! [A4: set_num,A: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ X4 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1295_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1296_finite__has__minimal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ X4 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1297_finite__list,axiom,
! [A4: set_VEBT_VEBT] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ? [Xs2: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_1298_finite__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_1299_finite__list,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ? [Xs2: list_int] :
( ( set_int2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_1300_finite__list,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ? [Xs2: list_complex] :
( ( set_complex2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_1301_finite__list,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ? [Xs2: list_P6011104703257516679at_nat] :
( ( set_Pr5648618587558075414at_nat @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_1302_unbounded__k__infinite,axiom,
! [K2: nat,S2: set_nat] :
( ! [M: nat] :
( ( ord_less_nat @ K2 @ M )
=> ? [N5: nat] :
( ( ord_less_nat @ M @ N5 )
& ( member_nat @ N5 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1303_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1304_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1305_empty__subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A4 ) ).
% empty_subsetI
thf(fact_1306_empty__subsetI,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).
% empty_subsetI
thf(fact_1307_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_1308_empty__subsetI,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).
% empty_subsetI
thf(fact_1309_subset__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% subset_empty
thf(fact_1310_subset__empty,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
= ( A4 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_1311_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1312_subset__empty,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
= ( A4 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_1313_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_1314_empty__iff,axiom,
! [C2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_1315_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_1316_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1317_empty__iff,axiom,
! [C2: int] :
~ ( member_int @ C2 @ bot_bot_set_int ) ).
% empty_iff
thf(fact_1318_all__not__in__conv,axiom,
! [A4: set_set_nat] :
( ( ! [X5: set_nat] :
~ ( member_set_nat @ X5 @ A4 ) )
= ( A4 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_1319_all__not__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ! [X5: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X5 @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_1320_all__not__in__conv,axiom,
! [A4: set_real] :
( ( ! [X5: real] :
~ ( member_real @ X5 @ A4 ) )
= ( A4 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_1321_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X5: nat] :
~ ( member_nat @ X5 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1322_all__not__in__conv,axiom,
! [A4: set_int] :
( ( ! [X5: int] :
~ ( member_int @ X5 @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_1323_Collect__empty__eq,axiom,
! [P2: list_nat > $o] :
( ( ( collect_list_nat @ P2 )
= bot_bot_set_list_nat )
= ( ! [X5: list_nat] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1324_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X5: set_nat] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1325_Collect__empty__eq,axiom,
! [P2: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P2 )
= bot_bo2099793752762293965at_nat )
= ( ! [X5: product_prod_nat_nat] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1326_Collect__empty__eq,axiom,
! [P2: real > $o] :
( ( ( collect_real @ P2 )
= bot_bot_set_real )
= ( ! [X5: real] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1327_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X5: nat] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1328_Collect__empty__eq,axiom,
! [P2: int > $o] :
( ( ( collect_int @ P2 )
= bot_bot_set_int )
= ( ! [X5: int] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1329_empty__Collect__eq,axiom,
! [P2: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P2 ) )
= ( ! [X5: list_nat] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1330_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X5: set_nat] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1331_empty__Collect__eq,axiom,
! [P2: product_prod_nat_nat > $o] :
( ( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ P2 ) )
= ( ! [X5: product_prod_nat_nat] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1332_empty__Collect__eq,axiom,
! [P2: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P2 ) )
= ( ! [X5: real] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1333_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X5: nat] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1334_empty__Collect__eq,axiom,
! [P2: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P2 ) )
= ( ! [X5: int] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1335_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X3: complex] :
( ( member_complex @ X3 @ S2 )
& ( ord_less_real @ ( F @ X3 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1336_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ord_less_real @ ( F @ X3 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1337_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ord_less_real @ ( F @ X3 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1338_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X3: int] :
( ( member_int @ X3 @ S2 )
& ( ord_less_real @ ( F @ X3 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1339_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X3: complex] :
( ( member_complex @ X3 @ S2 )
& ( ord_less_rat @ ( F @ X3 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1340_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ord_less_rat @ ( F @ X3 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1341_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ord_less_rat @ ( F @ X3 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1342_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X3: int] :
( ( member_int @ X3 @ S2 )
& ( ord_less_rat @ ( F @ X3 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1343_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X3: complex] :
( ( member_complex @ X3 @ S2 )
& ( ord_less_num @ ( F @ X3 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1344_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ord_less_num @ ( F @ X3 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1345_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1346_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1347_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1348_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1349_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1350_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1351_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > num] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1352_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > num] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1353_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1354_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1355_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
= ( P2 @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
=> ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1356_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_1357_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_1358_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_1359_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1360_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1361_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_1362_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_1363_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_1364_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1365_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1366_psubsetI,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_int @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_1367_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A7: set_int,B7: set_int] :
( ( ord_less_set_int @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1368_subset__psubset__trans,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_set_int @ B5 @ C4 )
=> ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_1369_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A7: set_int,B7: set_int] :
( ( ord_less_eq_set_int @ A7 @ B7 )
& ~ ( ord_less_eq_set_int @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1370_psubset__subset__trans,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C4 )
=> ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_1371_psubset__imp__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1372_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A7: set_int,B7: set_int] :
( ( ord_less_eq_set_int @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_1373_psubsetE,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_1374_psubsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C2 @ A4 )
=> ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1375_psubsetD,axiom,
! [A4: set_real,B5: set_real,C2: real] :
( ( ord_less_set_real @ A4 @ B5 )
=> ( ( member_real @ C2 @ A4 )
=> ( member_real @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1376_psubsetD,axiom,
! [A4: set_set_nat,B5: set_set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A4 @ B5 )
=> ( ( member_set_nat @ C2 @ A4 )
=> ( member_set_nat @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1377_psubsetD,axiom,
! [A4: set_nat,B5: set_nat,C2: nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C2 @ A4 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1378_psubsetD,axiom,
! [A4: set_int,B5: set_int,C2: int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( member_int @ C2 @ A4 )
=> ( member_int @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_1379_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_1380_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_1381_bot__set__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% bot_set_def
thf(fact_1382_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_1383_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1384_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_1385_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1386_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ ( ord_less_eq_rat @ A @ B )
| ~ ( ord_less_eq_rat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1387_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1388_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1389_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1390_verit__comp__simplify1_I2_J,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1391_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1392_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1393_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1394_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1395_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1396_verit__comp__simplify1_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1397_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1398_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1399_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1400_ex__in__conv,axiom,
! [A4: set_set_nat] :
( ( ? [X5: set_nat] : ( member_set_nat @ X5 @ A4 ) )
= ( A4 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_1401_ex__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 ) )
= ( A4 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_1402_ex__in__conv,axiom,
! [A4: set_real] :
( ( ? [X5: real] : ( member_real @ X5 @ A4 ) )
= ( A4 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_1403_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X5: nat] : ( member_nat @ X5 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1404_ex__in__conv,axiom,
! [A4: set_int] :
( ( ? [X5: int] : ( member_int @ X5 @ A4 ) )
= ( A4 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_1405_equals0I,axiom,
! [A4: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A4 )
=> ( A4 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_1406_equals0I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [Y3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y3 @ A4 )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_1407_equals0I,axiom,
! [A4: set_real] :
( ! [Y3: real] :
~ ( member_real @ Y3 @ A4 )
=> ( A4 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_1408_equals0I,axiom,
! [A4: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1409_equals0I,axiom,
! [A4: set_int] :
( ! [Y3: int] :
~ ( member_int @ Y3 @ A4 )
=> ( A4 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_1410_equals0D,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( A4 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1411_equals0D,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1412_equals0D,axiom,
! [A4: set_real,A: real] :
( ( A4 = bot_bot_set_real )
=> ~ ( member_real @ A @ A4 ) ) ).
% equals0D
thf(fact_1413_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1414_equals0D,axiom,
! [A4: set_int,A: int] :
( ( A4 = bot_bot_set_int )
=> ~ ( member_int @ A @ A4 ) ) ).
% equals0D
thf(fact_1415_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_1416_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_1417_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_1418_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1419_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_1420_not__psubset__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ).
% not_psubset_empty
thf(fact_1421_not__psubset__empty,axiom,
! [A4: set_real] :
~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_1422_not__psubset__empty,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1423_not__psubset__empty,axiom,
! [A4: set_int] :
~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_1424_verit__comp__simplify1_I3_J,axiom,
! [B2: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1425_verit__comp__simplify1_I3_J,axiom,
! [B2: rat,A2: rat] :
( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1426_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A2: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
= ( ord_less_num @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1427_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1428_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1429_subset__emptyI,axiom,
! [A4: set_set_nat] :
( ! [X4: set_nat] :
~ ( member_set_nat @ X4 @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_1430_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_1431_subset__emptyI,axiom,
! [A4: set_real] :
( ! [X4: real] :
~ ( member_real @ X4 @ A4 )
=> ( ord_less_eq_set_real @ A4 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_1432_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1433_subset__emptyI,axiom,
! [A4: set_int] :
( ! [X4: int] :
~ ( member_int @ X4 @ A4 )
=> ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1434_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_1435_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1436_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1437_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_1438_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1439_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1440_complete__interval,axiom,
! [A: real,B: real,P2: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: real] :
( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ C @ B )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ C ) )
=> ( P2 @ X3 ) )
& ! [D3: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_real @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1441_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C ) )
=> ( P2 @ X3 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1442_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C ) )
=> ( P2 @ X3 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1443_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1444_pinf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1445_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1446_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1447_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_1448_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1449_pinf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ord_less_eq_rat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1450_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ord_less_eq_num @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1451_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1452_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1453_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_1454_minf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ord_less_eq_rat @ X3 @ T ) ) ).
% minf(6)
thf(fact_1455_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ord_less_eq_num @ X3 @ T ) ) ).
% minf(6)
thf(fact_1456_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_1457_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_1458_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_1459_minf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ~ ( ord_less_eq_rat @ T @ X3 ) ) ).
% minf(8)
thf(fact_1460_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X3 ) ) ).
% minf(8)
thf(fact_1461_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_1462_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_1463_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1464_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_1465_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1466_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1467_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S3 ) ) ) ).
% deg_SUcn_Node
thf(fact_1468_count__notin,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( count_4203492906077236349at_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1469_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1470_count__notin,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1471_count__notin,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( count_list_VEBT_VEBT @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1472_count__notin,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( count_list_int @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1473_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_1474_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_1475_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1476_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1477_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1478_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1479_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_1480_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_1481_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1482_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1483_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1484_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1485_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1486_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1487_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va2: nat] :
( X
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_1488_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1489_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1490_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1491_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1492_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1493_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1494_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1495_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P2 @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P2 @ X4 @ Y3 )
=> ( P2 @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1496_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1497_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1498_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1499_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1500_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_1501_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1502_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1503_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1504_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_1505_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1506_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1507_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1508_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_1509_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1510_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1511_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_1512_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1513_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_1514_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1515_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K )
=> ( P2 @ I3 @ K ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1516_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1517_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1518_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1519_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_1520_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1521_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M: nat] :
( M7
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_1522_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_1523_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1524_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_1525_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1526_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1527_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1528_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1529_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1530_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1531_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_1532_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1533_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1534_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_1535_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1536_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_1537_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1538_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1539_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_1540_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1541_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1542_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1543_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1544_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_1545_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1546_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1547_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1548_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1549_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_1550_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1551_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1552_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1553_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1554_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_1555_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_1556_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1557_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1558_dec__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1559_inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1560_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_1561_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1562_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1563_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1564_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_1565_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1566_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1567_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1568_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1569_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1570_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1571_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1572_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1573_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_1574_minf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ~ ( ord_less_rat @ T @ X3 ) ) ).
% minf(7)
thf(fact_1575_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ~ ( ord_less_num @ T @ X3 ) ) ).
% minf(7)
thf(fact_1576_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_1577_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_1578_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_1579_minf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ord_less_rat @ X3 @ T ) ) ).
% minf(5)
thf(fact_1580_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ord_less_num @ X3 @ T ) ) ).
% minf(5)
thf(fact_1581_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_1582_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_1583_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1584_minf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1585_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1586_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1587_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1588_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1589_minf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1590_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1591_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1592_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1593_minf_I2_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1594_minf_I2_J,axiom,
! [P2: rat > $o,P5: rat > $o,Q: rat > $o,Q4: rat > $o] :
( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1595_minf_I2_J,axiom,
! [P2: num > $o,P5: num > $o,Q: num > $o,Q4: num > $o] :
( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1596_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1597_minf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1598_minf_I1_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1599_minf_I1_J,axiom,
! [P2: rat > $o,P5: rat > $o,Q: rat > $o,Q4: rat > $o] :
( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1600_minf_I1_J,axiom,
! [P2: num > $o,P5: num > $o,Q: num > $o,Q4: num > $o] :
( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1601_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1602_minf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1603_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_real @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1604_pinf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ord_less_rat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1605_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ord_less_num @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1606_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1607_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_1608_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_real @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1609_pinf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ~ ( ord_less_rat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1610_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ~ ( ord_less_num @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1611_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1612_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_1613_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1614_pinf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1615_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1616_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1617_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_1618_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1619_pinf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1620_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1621_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1622_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_1623_pinf_I2_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1624_pinf_I2_J,axiom,
! [P2: rat > $o,P5: rat > $o,Q: rat > $o,Q4: rat > $o] :
( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1625_pinf_I2_J,axiom,
! [P2: num > $o,P5: num > $o,Q: num > $o,Q4: num > $o] :
( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1626_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1627_pinf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1628_pinf_I1_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1629_pinf_I1_J,axiom,
! [P2: rat > $o,P5: rat > $o,Q: rat > $o,Q4: rat > $o] :
( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1630_pinf_I1_J,axiom,
! [P2: num > $o,P5: num > $o,Q: num > $o,Q4: num > $o] :
( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: num] :
! [X4: num] :
( ( ord_less_num @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1631_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1632_pinf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1633_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1634_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1635_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1636_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1637_ssubst__Pair__rhs,axiom,
! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1638_ssubst__Pair__rhs,axiom,
! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1639_ssubst__Pair__rhs,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1640_ssubst__Pair__rhs,axiom,
! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1641_ssubst__Pair__rhs,axiom,
! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1642_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_1643_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_1644_option_Osize__gen_I2_J,axiom,
! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1645_option_Osize__gen_I2_J,axiom,
! [X: num > nat,X2: num] :
( ( size_option_num @ X @ ( some_num @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1646_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X5: set_nat] : ( member_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1647_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1648_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X5: real] : ( member_real @ X5 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_1649_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X5: nat] : ( member_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1650_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X5: int] : ( member_int @ X5 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_1651_Collect__empty__eq__bot,axiom,
! [P2: list_nat > $o] :
( ( ( collect_list_nat @ P2 )
= bot_bot_set_list_nat )
= ( P2 = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1652_Collect__empty__eq__bot,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( P2 = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1653_Collect__empty__eq__bot,axiom,
! [P2: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P2 )
= bot_bo2099793752762293965at_nat )
= ( P2 = bot_bo482883023278783056_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1654_Collect__empty__eq__bot,axiom,
! [P2: real > $o] :
( ( ( collect_real @ P2 )
= bot_bot_set_real )
= ( P2 = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1655_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1656_Collect__empty__eq__bot,axiom,
! [P2: int > $o] :
( ( ( collect_int @ P2 )
= bot_bot_set_int )
= ( P2 = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1657_option_Osize_I4_J,axiom,
! [X2: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1658_option_Osize_I4_J,axiom,
! [X2: num] :
( ( size_size_option_num @ ( some_num @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1659_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1660_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1661__092_060open_062vebt__insert_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Ix_M_Amax_Ami_Ama_J_J_Adeg_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_J_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_092_060close_062,axiom,
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ xa @ ( ord_max_nat @ mi @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) ) ) ).
% \<open>vebt_insert (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (x, max mi ma)) deg (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)]) (if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary)\<close>
thf(fact_1662_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1663_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_1664_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z ) ) ).
% pair_less_iff1
thf(fact_1665_bit__split__inv,axiom,
! [X: nat,D: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
= X ) ).
% bit_split_inv
thf(fact_1666_list__update__overwrite,axiom,
! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 @ Y )
= ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ Y ) ) ).
% list_update_overwrite
thf(fact_1667_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
= ( ? [Y5: product_prod_nat_nat] :
( X
= ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_1668_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
= ( ? [Y5: num] :
( X
= ( some_num @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_1669_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( ! [Y5: product_prod_nat_nat] :
( X
!= ( some_P7363390416028606310at_nat @ Y5 ) ) )
= ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_1670_not__Some__eq,axiom,
! [X: option_num] :
( ( ! [Y5: num] :
( X
!= ( some_num @ Y5 ) ) )
= ( X = none_num ) ) ).
% not_Some_eq
thf(fact_1671_list__update__id,axiom,
! [Xs: list_nat,I2: nat] :
( ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_1672_list__update__id,axiom,
! [Xs: list_int,I2: nat] :
( ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_1673_list__update__id,axiom,
! [Xs: list_VEBT_VEBT,I2: nat] :
( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_1674_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs: list_nat,X: nat] :
( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1675_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs: list_int,X: int] :
( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
= ( nth_int @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1676_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1677_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1678_list__update__swap,axiom,
! [I2: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X8: vEBT_VEBT] :
( ( I2 != I5 )
=> ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I5 @ X8 )
= ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X8 ) @ I2 @ X ) ) ) ).
% list_update_swap
thf(fact_1679_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X )
!= ( size_s6755466524823107622T_VEBT @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1680_size__neq__size__imp__neq,axiom,
! [X: list_o,Y: list_o] :
( ( ( size_size_list_o @ X )
!= ( size_size_list_o @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1681_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1682_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1683_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_1684_option_Odistinct_I1_J,axiom,
! [X2: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_1685_option_Odistinct_I1_J,axiom,
! [X2: num] :
( none_num
!= ( some_num @ X2 ) ) ).
% option.distinct(1)
thf(fact_1686_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X2 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_1687_option_OdiscI,axiom,
! [Option: option_num,X2: num] :
( ( Option
= ( some_num @ X2 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_1688_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X22: product_prod_nat_nat] :
( Y
!= ( some_P7363390416028606310at_nat @ X22 ) ) ) ).
% option.exhaust
thf(fact_1689_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X22: num] :
( Y
!= ( some_num @ X22 ) ) ) ).
% option.exhaust
thf(fact_1690_split__option__ex,axiom,
( ( ^ [P3: option4927543243414619207at_nat > $o] :
? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option4927543243414619207at_nat > $o] :
( ( P4 @ none_P5556105721700978146at_nat )
| ? [X5: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_1691_split__option__ex,axiom,
( ( ^ [P3: option_num > $o] :
? [X6: option_num] : ( P3 @ X6 ) )
= ( ^ [P4: option_num > $o] :
( ( P4 @ none_num )
| ? [X5: num] : ( P4 @ ( some_num @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_1692_split__option__all,axiom,
( ( ^ [P3: option4927543243414619207at_nat > $o] :
! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option4927543243414619207at_nat > $o] :
( ( P4 @ none_P5556105721700978146at_nat )
& ! [X5: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_1693_split__option__all,axiom,
( ( ^ [P3: option_num > $o] :
! [X6: option_num] : ( P3 @ X6 ) )
= ( ^ [P4: option_num > $o] :
( ( P4 @ none_num )
& ! [X5: num] : ( P4 @ ( some_num @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_1694_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P2 @ X @ Y ) )
=> ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B3 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1695_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P2 @ X @ Y ) )
=> ( ! [A3: product_prod_nat_nat,B3: num] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y
= ( some_num @ B3 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1696_combine__options__cases,axiom,
! [X: option_num,P2: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P2 @ X @ Y ) )
=> ( ! [A3: num,B3: product_prod_nat_nat] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B3 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1697_combine__options__cases,axiom,
! [X: option_num,P2: option_num > option_num > $o,Y: option_num] :
( ( ( X = none_num )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P2 @ X @ Y ) )
=> ( ! [A3: num,B3: num] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y
= ( some_num @ B3 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_1698_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1699_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1700_option_Osize__gen_I1_J,axiom,
! [X: product_prod_nat_nat > nat] :
( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_1701_option_Osize__gen_I1_J,axiom,
! [X: num > nat] :
( ( size_option_num @ X @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_1702_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I2: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1703_set__update__subsetI,axiom,
! [Xs: list_real,A4: set_real,X: real,I2: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1704_set__update__subsetI,axiom,
! [Xs: list_set_nat,A4: set_set_nat,X: set_nat,I2: nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
=> ( ( member_set_nat @ X @ A4 )
=> ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1705_set__update__subsetI,axiom,
! [Xs: list_nat,A4: set_nat,X: nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1706_set__update__subsetI,axiom,
! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
=> ( ( member_VEBT_VEBT @ X @ A4 )
=> ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1707_set__update__subsetI,axiom,
! [Xs: list_int,A4: set_int,X: int,I2: nat] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I2 @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1708_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_1709_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_1710_subrelI,axiom,
! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ S ) )
=> ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_1711_subrelI,axiom,
! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ R2 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ S ) )
=> ( ord_le5997549366648089703at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_1712_subrelI,axiom,
! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
( ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S ) )
=> ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_1713_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X4: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_1714_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ! [X4: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_1715_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_1716_pair__lessI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1717_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList3: list_VEBT_VEBT,X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X5 @ N2 ) ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_1718_inthall,axiom,
! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o,N: nat] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1719_inthall,axiom,
! [Xs: list_real,P2: real > $o,N: nat] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( P2 @ ( nth_real @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1720_inthall,axiom,
! [Xs: list_set_nat,P2: set_nat > $o,N: nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P2 @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1721_inthall,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,N: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1722_inthall,axiom,
! [Xs: list_o,P2: $o > $o,N: nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( P2 @ ( nth_o @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1723_inthall,axiom,
! [Xs: list_nat,P2: nat > $o,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1724_inthall,axiom,
! [Xs: list_int,P2: int > $o,N: nat] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1725_pair__leqI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1726_pair__leqI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_1727__C4_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ na )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X3 @ na ) ) )
=> ( ( ord_less_nat @ mi @ X3 )
& ( ord_less_eq_nat @ X3 @ ma ) ) ) ) ) ) ).
% "4.hyps"(9)
thf(fact_1728_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_1729_vebt__insert_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_1730_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X4 ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1731_intind,axiom,
! [I2: nat,N: nat,P2: nat > $o,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_1732_intind,axiom,
! [I2: nat,N: nat,P2: int > $o,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth_int @ ( replicate_int @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_1733_intind,axiom,
! [I2: nat,N: nat,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_1734_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_1735_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_1736__C4_Ohyps_C_I2_J,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% "4.hyps"(2)
thf(fact_1737__C4_Ohyps_C_I8_J,axiom,
ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "4.hyps"(8)
thf(fact_1738__C4_Oprems_C_I1_J,axiom,
ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "4.prems"(1)
thf(fact_1739__C4_Oprems_C_I2_J,axiom,
ord_less_nat @ ya @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "4.prems"(2)
thf(fact_1740_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_1741__C4_OIH_C_I1_J,axiom,
! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( ( vEBT_invar_vebt @ X3 @ na )
& ! [Xa: nat] :
( ( ord_less_nat @ Xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
=> ! [Xb: nat] :
( ( ord_less_nat @ Xb @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ X3 @ Xa ) @ Xb )
=> ( ( vEBT_vebt_member @ X3 @ Xb )
| ( Xa = Xb ) ) ) ) ) ) ) ).
% "4.IH"(1)
thf(fact_1742__C4_Ohyps_C_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% "4.hyps"(5)
thf(fact_1743_high__bound__aux,axiom,
! [Ma: nat,N: nat,M2: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% high_bound_aux
thf(fact_1744_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_1745_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1746_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1747_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1748_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1749_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1750_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1751_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1752_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1753_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1754_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1755_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1756_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1757_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1758_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1759_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1760_add__numeral__left,axiom,
! [V: num,W2: num,Z: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1761_add__numeral__left,axiom,
! [V: num,W2: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1762_add__numeral__left,axiom,
! [V: num,W2: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1763_add__numeral__left,axiom,
! [V: num,W2: num,Z: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1764_add__numeral__left,axiom,
! [V: num,W2: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1765_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_1766_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_1767_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_1768_replicate__eq__replicate,axiom,
! [M2: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( replicate_VEBT_VEBT @ M2 @ X )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_1769_length__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1770_length__replicate,axiom,
! [N: nat,X: $o] :
( ( size_size_list_o @ ( replicate_o @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1771_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1772_length__replicate,axiom,
! [N: nat,X: int] :
( ( size_size_list_int @ ( replicate_int @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1773_length__list__update,axiom,
! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% length_list_update
thf(fact_1774_length__list__update,axiom,
! [Xs: list_o,I2: nat,X: $o] :
( ( size_size_list_o @ ( list_update_o @ Xs @ I2 @ X ) )
= ( size_size_list_o @ Xs ) ) ).
% length_list_update
thf(fact_1775_length__list__update,axiom,
! [Xs: list_nat,I2: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I2 @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_1776_length__list__update,axiom,
! [Xs: list_int,I2: nat,X: int] :
( ( size_size_list_int @ ( list_update_int @ Xs @ I2 @ X ) )
= ( size_size_list_int @ Xs ) ) ).
% length_list_update
thf(fact_1777_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_1778__C4_OIH_C_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ summary @ X ) @ Y )
=> ( ( vEBT_vebt_member @ summary @ Y )
| ( X = Y ) ) ) ) ) ).
% "4.IH"(2)
thf(fact_1779_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ V ) ) )
& ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1780_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1781_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1782_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1783_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1784_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(4)
thf(fact_1785_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(4)
thf(fact_1786_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(4)
thf(fact_1787_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(4)
thf(fact_1788_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(4)
thf(fact_1789_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(3)
thf(fact_1790_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(3)
thf(fact_1791_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(3)
thf(fact_1792_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(3)
thf(fact_1793_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(3)
thf(fact_1794_xyprop,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% xyprop
thf(fact_1795__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ ya @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
thf(fact_1796_Ball__set__replicate,axiom,
! [N: nat,A: int,P2: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1797_Ball__set__replicate,axiom,
! [N: nat,A: nat,P2: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1798_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1799_Bex__set__replicate,axiom,
! [N: nat,A: int,P2: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1800_Bex__set__replicate,axiom,
! [N: nat,A: nat,P2: nat > $o] :
( ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1801_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ? [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1802_in__set__replicate,axiom,
! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1803_in__set__replicate,axiom,
! [X: real,N: nat,Y: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1804_in__set__replicate,axiom,
! [X: set_nat,N: nat,Y: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1805_in__set__replicate,axiom,
! [X: int,N: nat,Y: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1806_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1807_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1808_list__update__beyond,axiom,
! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I2 )
=> ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1809_list__update__beyond,axiom,
! [Xs: list_o,I2: nat,X: $o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I2 )
=> ( ( list_update_o @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1810_list__update__beyond,axiom,
! [Xs: list_nat,I2: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I2 )
=> ( ( list_update_nat @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1811_list__update__beyond,axiom,
! [Xs: list_int,I2: nat,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I2 )
=> ( ( list_update_int @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1812_nth__replicate,axiom,
! [I2: nat,N: nat,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_1813_nth__replicate,axiom,
! [I2: nat,N: nat,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_1814_nth__replicate,axiom,
! [I2: nat,N: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_1815__C00_C,axiom,
( ( deg
= ( plus_plus_nat @ na @ m ) )
& ( ord_less_eq_nat @ zero_zero_nat @ na )
& ( na = m )
& ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg )
& ( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% "00"
thf(fact_1816_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1817_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1818_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1819_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1820_mimaxyprop,axiom,
( ~ ( ( xa = mi )
| ( xa = ma ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ mi @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ mi @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% mimaxyprop
thf(fact_1821_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1822_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1823_set__swap,axiom,
! [I2: nat,Xs: list_VEBT_VEBT,J: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
= ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1824_set__swap,axiom,
! [I2: nat,Xs: list_o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
=> ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I2 @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I2 ) ) )
= ( set_o2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1825_set__swap,axiom,
! [I2: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1826_set__swap,axiom,
! [I2: nat,Xs: list_int,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I2 ) ) )
= ( set_int2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1827_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1828_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_o] :
( ( size_size_list_o @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1829_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1830_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1831_neq__if__length__neq,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
!= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1832_neq__if__length__neq,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
!= ( size_size_list_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1833_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_1834_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1835_replicate__eqI,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= N )
=> ( ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1836_replicate__eqI,axiom,
! [Xs: list_real,N: nat,X: real] :
( ( ( size_size_list_real @ Xs )
= N )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_real @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1837_replicate__eqI,axiom,
! [Xs: list_set_nat,N: nat,X: set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= N )
=> ( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_set_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1838_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= N )
=> ( ! [Y3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1839_replicate__eqI,axiom,
! [Xs: list_o,N: nat,X: $o] :
( ( ( size_size_list_o @ Xs )
= N )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_o @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1840_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1841_replicate__eqI,axiom,
! [Xs: list_int,N: nat,X: int] :
( ( ( size_size_list_int @ Xs )
= N )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_int @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1842_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( X4 = X ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1843_replicate__length__same,axiom,
! [Xs: list_o,X: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( X4 = X ) )
=> ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1844_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( X4 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1845_replicate__length__same,axiom,
! [Xs: list_int,X: int] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( X4 = X ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1846_less__by__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_1847_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1848_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1849_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_1850_numeral__Bit0,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1851_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_1852_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1853_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_1854_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1855_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1856_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1857_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1858_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1859_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1860_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1861_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,D4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1862_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1863_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_1864_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_1865_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_1866_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1867_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_1868_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_1869_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_1870_finite__maxlen,axiom,
! [M5: set_list_VEBT_VEBT] :
( ( finite3004134309566078307T_VEBT @ M5 )
=> ? [N3: nat] :
! [X3: list_VEBT_VEBT] :
( ( member2936631157270082147T_VEBT @ X3 @ M5 )
=> ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1871_finite__maxlen,axiom,
! [M5: set_list_o] :
( ( finite_finite_list_o @ M5 )
=> ? [N3: nat] :
! [X3: list_o] :
( ( member_list_o @ X3 @ M5 )
=> ( ord_less_nat @ ( size_size_list_o @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1872_finite__maxlen,axiom,
! [M5: set_list_nat] :
( ( finite8100373058378681591st_nat @ M5 )
=> ? [N3: nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ M5 )
=> ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1873_finite__maxlen,axiom,
! [M5: set_list_int] :
( ( finite3922522038869484883st_int @ M5 )
=> ? [N3: nat] :
! [X3: list_int] :
( ( member_list_int @ X3 @ M5 )
=> ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_1874_length__induct,axiom,
! [P2: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1875_length__induct,axiom,
! [P2: list_o > $o,Xs: list_o] :
( ! [Xs2: list_o] :
( ! [Ys2: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1876_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1877_length__induct,axiom,
! [P2: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1878_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_1879_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_1880_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_1881_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_1882_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_1883_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_1884_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1885_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_1886_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_1887_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_1888_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1889_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_1890_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_1891_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_1892_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_1893_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1894_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_1895_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_1896_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1897_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1898_zero__less__numeral,axiom,
! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_less_numeral
thf(fact_1899_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_1900_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_1901_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1902_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1903_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_less_zero
thf(fact_1904_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_1905_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : Y4 = Z2 )
= ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( ( nth_VEBT_VEBT @ Xs3 @ I4 )
= ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1906_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_o,Z2: list_o] : Y4 = Z2 )
= ( ^ [Xs3: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= ( size_size_list_o @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs3 ) )
=> ( ( nth_o @ Xs3 @ I4 )
= ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1907_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2 )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1908_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z2: list_int] : Y4 = Z2 )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I4 )
= ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1909_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > vEBT_VEBT > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X9: vEBT_VEBT] : ( P2 @ I4 @ X9 ) ) )
= ( ? [Xs3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth_VEBT_VEBT @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1910_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > $o > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X9: $o] : ( P2 @ I4 @ X9 ) ) )
= ( ? [Xs3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth_o @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1911_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X9: nat] : ( P2 @ I4 @ X9 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1912_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > int > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X9: int] : ( P2 @ I4 @ X9 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1913_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I3 )
= ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1914_nth__equalityI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I3 )
= ( nth_o @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1915_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1916_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I3 )
= ( nth_int @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1917_set__encode__eq,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_1918_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1919_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_1920_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A3: $o,B3: $o] :
( A1
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( A22
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1921_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A5: $o,B4: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B4 ) )
& ( A23
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A23
= ( plus_plus_nat @ N2 @ N2 ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A23
= ( plus_plus_nat @ N2 @ N2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1922_count__le__length,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% count_le_length
thf(fact_1923_count__le__length,axiom,
! [Xs: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X ) @ ( size_size_list_o @ Xs ) ) ).
% count_le_length
thf(fact_1924_count__le__length,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_1925_count__le__length,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X ) @ ( size_size_list_int @ Xs ) ) ).
% count_le_length
thf(fact_1926_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_1927_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1928_invar__vebt_Ointros_I1_J,axiom,
! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% invar_vebt.intros(1)
thf(fact_1929_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1930_length__pos__if__in__set,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1931_length__pos__if__in__set,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1932_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1933_length__pos__if__in__set,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1934_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1935_length__pos__if__in__set,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1936_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1937_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_1938_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1939_nth__mem,axiom,
! [N: nat,Xs: list_set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1940_nth__mem,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1941_nth__mem,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1942_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1943_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1944_list__ball__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1945_list__ball__nth,axiom,
! [N: nat,Xs: list_o,P2: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_o @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1946_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1947_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P2: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1948_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1949_in__set__conv__nth,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1950_in__set__conv__nth,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1951_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( ( nth_VEBT_VEBT @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1952_in__set__conv__nth,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
& ( ( nth_o @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1953_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1954_in__set__conv__nth,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1955_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1956_all__nth__imp__all__set,axiom,
! [Xs: list_real,P2: real > $o,X: real] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
=> ( P2 @ ( nth_real @ Xs @ I3 ) ) )
=> ( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1957_all__nth__imp__all__set,axiom,
! [Xs: list_set_nat,P2: set_nat > $o,X: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P2 @ ( nth_set_nat @ Xs @ I3 ) ) )
=> ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1958_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
=> ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1959_all__nth__imp__all__set,axiom,
! [Xs: list_o,P2: $o > $o,X: $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( P2 @ ( nth_o @ Xs @ I3 ) ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1960_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1961_all__nth__imp__all__set,axiom,
! [Xs: list_int,P2: int > $o,X: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ I3 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1962_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1963_all__set__conv__all__nth,axiom,
! [Xs: list_o,P2: $o > $o] :
( ( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( P2 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1964_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1965_all__set__conv__all__nth,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1966_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_1967_set__update__memI,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1968_set__update__memI,axiom,
! [N: nat,Xs: list_real,X: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1969_set__update__memI,axiom,
! [N: nat,Xs: list_set_nat,X: set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1970_set__update__memI,axiom,
! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1971_set__update__memI,axiom,
! [N: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1972_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1973_set__update__memI,axiom,
! [N: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1974_nth__list__update,axiom,
! [I2: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1975_nth__list__update,axiom,
! [I2: nat,Xs: list_o,X: $o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ J )
= ( ( ( I2 = J )
=> X )
& ( ( I2 != J )
=> ( nth_o @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1976_nth__list__update,axiom,
! [I2: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1977_nth__list__update,axiom,
! [I2: nat,Xs: list_int,J: nat,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
= ( nth_int @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1978_list__update__same__conv,axiom,
! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
= Xs )
= ( ( nth_VEBT_VEBT @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1979_list__update__same__conv,axiom,
! [I2: nat,Xs: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( ( list_update_o @ Xs @ I2 @ X )
= Xs )
= ( ( nth_o @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1980_list__update__same__conv,axiom,
! [I2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I2 @ X )
= Xs )
= ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1981_list__update__same__conv,axiom,
! [I2: nat,Xs: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( ( list_update_int @ Xs @ I2 @ X )
= Xs )
= ( ( nth_int @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1982_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1983_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1984_set__encode__inf,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_1985_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1986_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1987_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1988_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1989_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_1990_zero__less__power2,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_power2
thf(fact_1991_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_1992_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1993_power2__eq__iff__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1994_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1995_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1996_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_1997_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
thf(fact_1998_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1999_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ X @ Mi )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_2000_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ Mi @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_2001_zero__eq__power2,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_2002_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_2003_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_2004_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_2005_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_2006_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2007_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2008_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2009_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2010_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( power_power_rat @ A @ N )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2011_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2012_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2013_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2014_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2015_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less_nat @ X @ Ma )
& ( ord_less_nat @ Mi @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_2016_pow__sum,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% pow_sum
thf(fact_2017_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X5: nat,N2: nat] : ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_2018_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_2019_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_2020_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_2021_divide__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C2 )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2022_divide__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C2 )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2023_divide__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2024_divide__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C2 @ A )
= ( divide1717551699836669952omplex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2025_divide__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C2 @ A )
= ( divide_divide_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2026_divide__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( divide_divide_real @ C2 @ A )
= ( divide_divide_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2027_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_2028_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_2029_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_2030_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_2031_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_2032_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_2033_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_2034_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_2035_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_2036_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_2037_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_2038_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_2039_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_2040_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_2041_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_2042_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_2043_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_2044_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_2045_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_2046_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_2047_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_2048_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_2049_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_2050_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2051_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2052_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2053_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2054_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_2055_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_2056_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_2057_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_2058_add__divide__distrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_2059_add__divide__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_2060_divide__right__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2061_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( divide_divide_rat @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2062_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2063_divide__nonpos__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2064_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2065_divide__nonpos__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2066_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2067_divide__nonneg__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2068_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2069_divide__nonneg__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2070_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2071_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2072_divide__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_2073_divide__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_2074_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2075_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2076_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2077_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2078_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2079_divide__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2080_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2081_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2082_divide__less__cancel,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) )
& ( C2 != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_2083_divide__less__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_2084_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2085_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2086_divide__pos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_2087_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_2088_divide__pos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_2089_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_2090_divide__neg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_2091_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_2092_divide__neg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_2093_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_2094_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_2095_divide__nonpos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_2096_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2097_divide__nonpos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2098_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2099_divide__nonneg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2100_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_2101_divide__nonneg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_2102_divide__le__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2103_divide__le__cancel,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2104_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2105_frac__less2,axiom,
! [X: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2106_frac__less,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2107_frac__less,axiom,
! [X: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2108_frac__le,axiom,
! [Y: real,X: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2109_frac__le,axiom,
! [Y: rat,X: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2110_field__sum__of__halves,axiom,
! [X: rat] :
( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_2111_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_2112_half__gt__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2113_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2114_half__gt__zero__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% half_gt_zero_iff
thf(fact_2115_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_2116_field__less__half__sum,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_2117_field__less__half__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_2118_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_2119_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_2120_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_2121_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_2122_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_2123_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2124_zero__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2125_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2126_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2127_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2128_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2129_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2130_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2131_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2132_zero__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2133_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2134_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2135_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_2136_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2137_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2138_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2139_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2140_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2141_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2142_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2143_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2144_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2145_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N ) )
= ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2146_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2147_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2148_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_2149_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_2150_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_2151_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_2152_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_2153_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_2154_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_2155_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2156_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2157_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2158_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2159_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2160_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2161_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2162_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2163_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_2164_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_2165_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_2166_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_2167_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_2168_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_2169_power2__nat__le__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_2170_power2__nat__le__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_2171_self__le__ge2__pow,axiom,
! [K2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K2 @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_2172_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2173_zero__le__power2,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2174_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2175_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2176_power2__eq__imp__eq,axiom,
! [X: rat,Y: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2177_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2178_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2179_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2180_power2__le__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2181_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2182_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2183_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2184_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2185_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2186_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2187_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_2188_power2__less__0,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% power2_less_0
thf(fact_2189_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_2190_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2191_power2__less__imp__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2192_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2193_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2194_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2195_sum__power2__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2196_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2197_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2198_sum__power2__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2199_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2200_sum__power2__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2201_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2202_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2203_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_2204_not__sum__power2__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_2205_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_2206_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_2207_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_2208_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_2209_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_2210_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_2211_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_2212_set__n__deg__not__0,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_2213_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2214_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2215_div__exp__eq,axiom,
! [A: nat,M2: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% div_exp_eq
thf(fact_2216_div__exp__eq,axiom,
! [A: int,M2: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% div_exp_eq
thf(fact_2217_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A5: $o,B4: $o] :
( T
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg1Leaf
thf(fact_2218_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A3: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% deg_1_Leaf
thf(fact_2219_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A3: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% deg_1_Leafy
thf(fact_2220_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_2221_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_2222_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_2223_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_2224_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_2225_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_2226_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_2227_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_2228_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_2229_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_2230_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_2231_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_2232_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2233_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2234_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2235_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_2236_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_2237_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_2238_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_2239_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_2240_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2241_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2242_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2243_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_2244_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_2245_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_2246_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_2247_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_2248_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_2249_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2250_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2251_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2252_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2253_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_2254_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_2255_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_2256_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_2257_power__inject__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_2258_power__inject__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M2 )
= ( power_power_rat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_2259_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_2260_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_2261_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_2262_max__0__1_I2_J,axiom,
( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
= one_one_rat ) ).
% max_0_1(2)
thf(fact_2263_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_2264_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_2265_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_2266_max__0__1_I1_J,axiom,
( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
= one_one_rat ) ).
% max_0_1(1)
thf(fact_2267_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_2268_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_2269_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(6)
thf(fact_2270_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(6)
thf(fact_2271_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(6)
thf(fact_2272_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(6)
thf(fact_2273_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(6)
thf(fact_2274_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(5)
thf(fact_2275_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(5)
thf(fact_2276_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(5)
thf(fact_2277_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(5)
thf(fact_2278_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(5)
thf(fact_2279_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_2280_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_2281_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_2282_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2283_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2284_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_2285_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_2286_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2287_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2288_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2289_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2290_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2291_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2292_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2293_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2294_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2295_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2296_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_2297_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_2298_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_2299_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_2300_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2301_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2302_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2303_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2304_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2305_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2306_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2307_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2308_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2309_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2310_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2311_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2312_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2313_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_2314_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2315_power__strict__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2316_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2317_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2318_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2319_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2320_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2321_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2322_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2323_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2324_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2325_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2326_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2327_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_2328_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2329_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2330_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2331_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2332_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2333_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_2334_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2335_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2336_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2337_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2338_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_2339_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2340_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2341_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2342_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2343_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_2344_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_2345_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_2346_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_2347_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_2348_power__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2349_power__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2350_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2351_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2352_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_2353_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_2354_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_2355_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_2356_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_2357_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_2358_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_2359_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_2360_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_2361_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_2362_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_2363_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_2364_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_2365_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_2366_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_2367_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_2368_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_2369_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_2370_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_2371_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_2372_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_2373_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_2374_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_2375_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_2376_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_2377_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_2378_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2379_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2380_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2381_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2382_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_2383_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_2384_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_2385_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_2386_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_2387_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_2388_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_2389_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_2390_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_2391_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_2392_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_2393_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_2394_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_2395_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_2396_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_2397_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_2398_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_2399_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_2400_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_2401_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_2402_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_2403_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_2404_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_2405_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_2406_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% not_numeral_less_one
thf(fact_2407_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_2408_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_2409_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_2410_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_2411_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_2412_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_2413_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_2414_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_2415_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_2416_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2417_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2418_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2419_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% one_plus_numeral_commute
thf(fact_2420_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_2421_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_2422_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_2423_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_2424_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_2425_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_2426_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_2427_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_2428_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_2429_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_2430_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_2431_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_2432_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_2433_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_2434_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_2435_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_2436_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_2437_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_2438_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_2439_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_2440_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
= ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_2441_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_2442_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_2443_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_2444_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_2445_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_2446_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_2447_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_2448_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_2449_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_2450_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_2451_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2452_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2453_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_2454_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_2455_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_2456_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_2457_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_2458_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_2459_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_2460_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_2461_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_2462_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2463_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2464_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2465_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2466_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_2467_power__less__imp__less__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_2468_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_2469_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_2470_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_2471_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: rat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_2472_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_2473_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_2474_power__increasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_2475_power__increasing,axiom,
! [N: nat,N6: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_2476_power__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_2477_power__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_2478_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_2479_vebt__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
= ( ( ( X = zero_zero_nat )
=> A )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B )
& ( X = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_2480_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A: $o,B: $o] :
( ( ( X = zero_zero_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ $true @ B ) ) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ $true ) ) )
& ( ( X != one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_2481_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
= ( ( ( X = zero_zero_nat )
=> A )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B )
& ( X = one_one_nat ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_2482_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_2483_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_2484_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2485_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2486_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2487_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2488_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2489_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2490_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_2491_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_2492_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_2493_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_2494_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2495_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: rat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2496_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2497_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2498_power__decreasing,axiom,
! [N: nat,N6: nat,A: real] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2499_power__decreasing,axiom,
! [N: nat,N6: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2500_power__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2501_power__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2502_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2503_power__le__imp__le__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2504_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2505_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2506_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2507_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2508_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2509_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2510_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2511_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2512_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2513_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2514_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_2515_div__le__mono,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_2516_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_2517_ex__power__ivl2,axiom,
! [B: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2518_ex__power__ivl1,axiom,
! [B: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2519_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_2520_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_2521_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_2522_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_2523_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2524_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2525_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2526_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2527_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_2528_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_2529_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_2530_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_2531_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_2532_enat__ord__number_I2_J,axiom,
! [M2: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_2533_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_2534_discrete,axiom,
( ord_less_nat
= ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_2535_discrete,axiom,
( ord_less_int
= ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_2536_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2537_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2538_div__positive,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2539_div__positive,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2540_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_2541_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_2542_mult__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( ( times_times_complex @ A @ C2 )
= ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2543_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2544_mult__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ( times_times_rat @ A @ C2 )
= ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2545_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2546_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2547_mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C2 @ A )
= ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2548_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2549_mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C2 @ A )
= ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2550_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2551_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2552_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_2553_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_2554_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_2555_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_2556_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_2557_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_2558_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_2559_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_2560_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_2561_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_2562_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_2563_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_2564_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_2565_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_2566_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_2567_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_2568_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_2569_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_2570_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_2571_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_2572_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_2573_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_2574_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_2575_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_2576_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_2577_times__divide__eq__left,axiom,
! [B: complex,C2: complex,A: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C2 ) @ A )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_2578_times__divide__eq__left,axiom,
! [B: rat,C2: rat,A: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_2579_times__divide__eq__left,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_2580_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C2 )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_2581_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C2 )
= ( divide_divide_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_2582_divide__divide__eq__left,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_2583_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C2: complex] :
( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_2584_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_2585_divide__divide__eq__right,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_2586_times__divide__eq__right,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_2587_times__divide__eq__right,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_2588_times__divide__eq__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_2589_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_2590_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_2591_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_2592_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_2593_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_2594_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_2595_mult__cancel__right2,axiom,
! [A: complex,C2: complex] :
( ( ( times_times_complex @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_2596_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_2597_mult__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ( times_times_rat @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_2598_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_2599_mult__cancel__right1,axiom,
! [C2: complex,B: complex] :
( ( C2
= ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_2600_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_2601_mult__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( C2
= ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_2602_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_2603_mult__cancel__left2,axiom,
! [C2: complex,A: complex] :
( ( ( times_times_complex @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_2604_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_2605_mult__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ( times_times_rat @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_2606_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_2607_mult__cancel__left1,axiom,
! [C2: complex,B: complex] :
( ( C2
= ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_2608_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_2609_mult__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( C2
= ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_2610_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_2611_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2612_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2613_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2614_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2615_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2616_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2617_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2618_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2619_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2620_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2621_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2622_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2623_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2624_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2625_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2626_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ B @ C2 ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2627_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2628_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2629_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2630_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2631_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2632_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2633_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2634_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2635_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2636_mult__divide__mult__cancel__left__if,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( C2 = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= zero_zero_complex ) )
& ( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2637_mult__divide__mult__cancel__left__if,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( C2 = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= zero_zero_rat ) )
& ( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2638_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2639_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2640_div__mult__mult1__if,axiom,
! [C2: int,A: int,B: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2641_div__mult__mult2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2642_div__mult__mult2,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2643_div__mult__mult1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2644_div__mult__mult1,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2645_distrib__right__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2646_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2647_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2648_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2649_distrib__right__numeral,axiom,
! [A: extended_enat,B: extended_enat,V: num] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2650_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2651_distrib__left__numeral,axiom,
! [V: num,B: complex,C2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2652_distrib__left__numeral,axiom,
! [V: num,B: rat,C2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2653_distrib__left__numeral,axiom,
! [V: num,B: nat,C2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2654_distrib__left__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2655_distrib__left__numeral,axiom,
! [V: num,B: extended_enat,C2: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C2 ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2656_distrib__left__numeral,axiom,
! [V: num,B: real,C2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2657_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_2658_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_2659_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_2660_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_2661_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_2662_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_2663_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_2664_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_2665_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_2666_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2667_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2668_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2669_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2670_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2671_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2672_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2673_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2674_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2675_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2676_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2677_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2678_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2679_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2680_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2681_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2682_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2683_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2684_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2685_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2686_div__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2687_div__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2688_div__mult__self3,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2689_div__mult__self3,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2690_div__mult__self2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2691_div__mult__self2,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2692_div__mult__self1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2693_div__mult__self1,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2694_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_2695_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_2696_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_2697_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_2698_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_2699_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_2700_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_2701_zdiv__mono1,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_2702_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_2703_zdiv__eq__0__iff,axiom,
! [I2: int,K2: int] :
( ( ( divide_divide_int @ I2 @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K2 ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K2 @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_2704_zdiv__mono1__neg,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_2705_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_2706_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_2707_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_2708_div__positive__int,axiom,
! [L: int,K2: int] :
( ( ord_less_eq_int @ L @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) ) ) ) ).
% div_positive_int
thf(fact_2709_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_2710_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_2711_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K2 ) )
= ( ord_less_eq_int @ K2 @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_2712_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_2713_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_2714_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_2715_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_2716_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_2717_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_2718_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_2719_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K2 )
=> ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_2720_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_2721_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_2722_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_2723_int__gr__induct,axiom,
! [K2: int,I2: int,P2: int > $o] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_2724_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C2 )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2725_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2726_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2727_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2728_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2729_mult_Oassoc,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C2 )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2730_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2731_mult_Oassoc,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2732_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2733_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2734_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A5: complex,B4: complex] : ( times_times_complex @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2735_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2736_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2737_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2738_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2739_mult_Oleft__commute,axiom,
! [B: complex,A: complex,C2: complex] :
( ( times_times_complex @ B @ ( times_times_complex @ A @ C2 ) )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2740_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2741_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C2: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C2 ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2742_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2743_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2744_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_2745_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_2746_enat__less__induct,axiom,
! [P2: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_le72135733267957522d_enat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% enat_less_induct
thf(fact_2747_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_2748_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_2749_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_2750_mult__right__cancel,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C2 )
= ( times_times_complex @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2751_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2752_mult__right__cancel,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C2 )
= ( times_times_rat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2753_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2754_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2755_mult__left__cancel,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A )
= ( times_times_complex @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2756_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2757_mult__left__cancel,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ C2 @ A )
= ( times_times_rat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2758_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2759_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2760_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_2761_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_2762_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_2763_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_2764_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_2765_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_2766_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_2767_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_2768_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_2769_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_2770_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_2771_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_2772_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_2773_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_2774_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_2775_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2776_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2777_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2778_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2779_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2780_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_2781_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_2782_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_2783_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_2784_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_2785_crossproduct__noteq,axiom,
! [A: complex,B: complex,C2: complex,D: complex] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ D ) )
!= ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2786_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2787_crossproduct__noteq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) )
!= ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2788_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2789_crossproduct__noteq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2790_crossproduct__eq,axiom,
! [W2: complex,Y: complex,X: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y ) @ ( times_times_complex @ X @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W2 @ Z ) @ ( times_times_complex @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_2791_crossproduct__eq,axiom,
! [W2: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W2 @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_2792_crossproduct__eq,axiom,
! [W2: rat,Y: rat,X: rat,Z: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z ) )
= ( plus_plus_rat @ ( times_times_rat @ W2 @ Z ) @ ( times_times_rat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_2793_crossproduct__eq,axiom,
! [W2: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_2794_crossproduct__eq,axiom,
! [W2: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_2795_combine__common__factor,axiom,
! [A: complex,E2: complex,B: complex,C2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2796_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2797_combine__common__factor,axiom,
! [A: rat,E2: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2798_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2799_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2800_distrib__right,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2801_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2802_distrib__right,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2803_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2804_distrib__right,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2805_distrib__left,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2806_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2807_distrib__left,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2808_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2809_distrib__left,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2810_comm__semiring__class_Odistrib,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2811_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2812_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2813_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2814_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2815_ring__class_Oring__distribs_I1_J,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2816_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2817_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2818_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2819_ring__class_Oring__distribs_I2_J,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2820_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2821_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2822_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2823_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W2: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_2824_times__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W2: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_2825_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_2826_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_2827_divide__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W2: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_2828_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_2829_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C2 )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C2 @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_2830_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C2 )
= ( divide_divide_rat @ A @ ( times_times_rat @ C2 @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_2831_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ C2 @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_2832_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_2833_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_2834_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_2835_mult__le__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_2836_mult__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_2837_mult__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_2838_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_2839_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_2840_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_2841_add__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_2842_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_2843_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_2844_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_2845_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_2846_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M2 @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_2847_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2848_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2849_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2850_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2851_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2852_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2853_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2854_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2855_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2856_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2857_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2858_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2859_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2860_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2861_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2862_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2863_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2864_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2865_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2866_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2867_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2868_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2869_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2870_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2871_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2872_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_2873_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_2874_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_2875_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_2876_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2877_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2878_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2879_mult__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2880_mult__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2881_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2882_mult__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2883_mult__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2884_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2885_mult__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2886_mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2887_mult__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2888_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2889_mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2890_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2891_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2892_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2893_mult__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2894_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2895_mult__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2896_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2897_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2898_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2899_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_2900_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_2901_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_2902_mult__mono_H,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2903_mult__mono_H,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2904_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2905_mult__mono_H,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2906_mult__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2907_mult__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2908_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2909_mult__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2910_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2911_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2912_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2913_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2914_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2915_mult__less__cancel__right__disj,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C2 @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2916_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2917_mult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2918_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2919_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2920_mult__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2921_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2922_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2923_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2924_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2925_mult__less__cancel__left__disj,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C2 @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2926_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2927_mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2928_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2929_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2930_mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2931_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2932_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2933_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2934_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2935_mult__less__cancel__left__pos,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2936_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2937_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2938_mult__less__cancel__left__neg,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2939_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2940_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2941_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2942_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2943_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2944_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2945_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2946_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2947_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2948_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2949_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2950_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2951_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_2952_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_2953_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_2954_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_2955_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2956_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2957_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2958_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2959_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_2960_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_2961_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_2962_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_2963_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_2964_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_2965_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_2966_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_2967_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2968_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2969_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2970_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_2971_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_2972_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_2973_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2974_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2975_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2976_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C2: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C2 ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2977_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C2: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2978_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C2: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C2 ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2979_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C2: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2980_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C2: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C2 ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2981_less__1__mult,axiom,
! [M2: real,N: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_2982_less__1__mult,axiom,
! [M2: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M2 )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_2983_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_2984_less__1__mult,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_2985_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W2 @ Z ) )
= ( ( times_times_complex @ X @ Z )
= ( times_times_complex @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2986_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y )
= ( divide_divide_rat @ W2 @ Z ) )
= ( ( times_times_rat @ X @ Z )
= ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2987_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W2 @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2988_divide__eq__eq,axiom,
! [B: complex,C2: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_2989_divide__eq__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_2990_divide__eq__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ( divide_divide_real @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_2991_eq__divide__eq,axiom,
! [A: complex,B: complex,C2: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_2992_eq__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_2993_eq__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_2994_divide__eq__imp,axiom,
! [C2: complex,B: complex,A: complex] :
( ( C2 != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C2 ) )
=> ( ( divide1717551699836669952omplex @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2995_divide__eq__imp,axiom,
! [C2: rat,B: rat,A: rat] :
( ( C2 != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C2 ) )
=> ( ( divide_divide_rat @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2996_divide__eq__imp,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2997_eq__divide__imp,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C2 )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_2998_eq__divide__imp,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C2 )
= B )
=> ( A
= ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_2999_eq__divide__imp,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B )
=> ( A
= ( divide_divide_real @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_3000_nonzero__divide__eq__eq,axiom,
! [C2: complex,B: complex,A: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C2 )
= A )
= ( B
= ( times_times_complex @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3001_nonzero__divide__eq__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C2 )
= A )
= ( B
= ( times_times_rat @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3002_nonzero__divide__eq__eq,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C2 )
= A )
= ( B
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3003_nonzero__eq__divide__eq,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( times_times_complex @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3004_nonzero__eq__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C2 ) )
= ( ( times_times_rat @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3005_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3006_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_3007_power__add,axiom,
! [A: complex,M2: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_3008_power__add,axiom,
! [A: real,M2: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_3009_power__add,axiom,
! [A: rat,M2: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_3010_power__add,axiom,
! [A: nat,M2: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_3011_power__add,axiom,
! [A: int,M2: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_3012_mult__less__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_3013_mult__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_3014_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_3015_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_3016_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_3017_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_3018_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_3019_less__mult__imp__div__less,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_3020_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_3021_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_3022_mult__less__le__imp__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3023_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3024_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3025_mult__less__le__imp__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3026_mult__le__less__imp__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3027_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3028_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3029_mult__le__less__imp__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3030_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3031_mult__right__le__imp__le,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3032_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3033_mult__right__le__imp__le,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3034_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3035_mult__left__le__imp__le,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3036_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3037_mult__left__le__imp__le,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3038_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3039_mult__le__cancel__left__pos,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3040_mult__le__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3041_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3042_mult__le__cancel__left__neg,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3043_mult__le__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3044_mult__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3045_mult__less__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3046_mult__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3047_mult__strict__mono_H,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3048_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3049_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3050_mult__strict__mono_H,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3051_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3052_mult__right__less__imp__less,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3053_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3054_mult__right__less__imp__less,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3055_mult__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3056_mult__less__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3057_mult__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3058_mult__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3059_mult__strict__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3060_mult__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3061_mult__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3062_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3063_mult__left__less__imp__less,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3064_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3065_mult__left__less__imp__less,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3066_mult__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3067_mult__le__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3068_mult__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3069_mult__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3070_mult__le__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3071_mult__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3072_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3073_mult__left__le,axiom,
! [C2: rat,A: rat] :
( ( ord_less_eq_rat @ C2 @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3074_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3075_mult__left__le,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3076_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_3077_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_3078_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_3079_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_3080_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3081_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3082_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3083_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3084_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3085_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3086_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3087_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3088_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3089_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3090_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3091_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3092_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3093_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3094_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3095_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_3096_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_3097_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_3098_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_3099_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_3100_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_3101_divide__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_3102_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_3103_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_3104_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_3105_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_3106_pos__less__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_3107_pos__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_3108_pos__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_3109_pos__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_3110_neg__less__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_3111_neg__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_3112_neg__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_3113_neg__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_3114_less__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_3115_less__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_3116_divide__less__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_3117_divide__less__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_3118_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C2: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= ( numera6690914467698888265omplex @ W2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3119_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C2 )
= ( numeral_numeral_rat @ W2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3120_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ( divide_divide_real @ B @ C2 )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3121_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: complex,C2: complex] :
( ( ( numera6690914467698888265omplex @ W2 )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3122_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ( numeral_numeral_rat @ W2 )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3123_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3124_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_3125_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_3126_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_3127_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_3128_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_3129_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_3130_add__num__frac,axiom,
! [Y: complex,Z: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3131_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3132_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3133_add__frac__num,axiom,
! [Y: complex,X: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3134_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3135_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3136_add__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_3137_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_3138_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_3139_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3140_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3141_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3142_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3143_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3144_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3145_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3146_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3147_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3148_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3149_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3150_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3151_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3152_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3153_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_3154_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_3155_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_3156_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_3157_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q2 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_3158_nat__mult__div__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_3159_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_3160_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3161_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z3 )
=> ( ( ord_less_rat @ Z3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3162_mult__less__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3163_mult__less__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3164_mult__less__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3165_mult__less__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3166_mult__less__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3167_mult__less__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3168_mult__less__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3169_mult__less__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3170_mult__less__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3171_mult__less__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3172_mult__less__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3173_mult__less__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3174_mult__le__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3175_mult__le__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3176_mult__le__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3177_mult__le__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3178_mult__le__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3179_mult__le__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3180_mult__le__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3181_mult__le__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3182_mult__le__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3183_mult__le__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3184_mult__le__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3185_mult__le__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3186_divide__left__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_3187_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_3188_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_3189_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_3190_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_3191_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_3192_pos__le__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_3193_pos__le__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_3194_pos__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_3195_pos__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_3196_neg__le__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_3197_neg__le__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_3198_neg__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_3199_neg__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_3200_divide__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_3201_divide__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_3202_le__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_3203_le__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_3204_divide__le__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_3205_divide__le__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_3206_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3207_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3208_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3209_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_3210_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_3211_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_3212_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_3213_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3214_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3215_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3216_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3217_mult__2,axiom,
! [Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2
thf(fact_3218_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_3219_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_3220_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_3221_mult__2,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2
thf(fact_3222_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_3223_mult__2__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3224_mult__2__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3225_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3226_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3227_mult__2__right,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3228_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_3229_left__add__twice,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3230_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3231_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3232_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3233_left__add__twice,axiom,
! [A: extended_enat,B: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3234_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3235_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M2 )
=> ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M2 @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_3236_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_3237_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_3238_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_3239_split__div,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
=> ( P2 @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_3240_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3241_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A )
=> ( ( ord_less_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3242_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3243_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3244_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3245_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3246_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3247_sum__squares__bound,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_3248_sum__squares__bound,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_3249_split__div_H,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P2 @ zero_zero_nat ) )
| ? [Q5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M2 )
& ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
& ( P2 @ Q5 ) ) ) ) ).
% split_div'
thf(fact_3250_power2__sum,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3251_power2__sum,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3252_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3253_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3254_power2__sum,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3255_power2__sum,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_3256_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3257_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3258_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3259_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_3260_arith__geo__mean,axiom,
! [U: real,X: real,Y: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3261_arith__geo__mean,axiom,
! [U: rat,X: rat,Y: rat] :
( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X @ Y ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3262_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_3263_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N )
= A ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_3264_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_3265_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3266_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3267_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3268_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_3269_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_3270_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_3271_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3272_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3273_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3274_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3275_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3276_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3277_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3278_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3279_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3280_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3281_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_3282_divides__aux__eq,axiom,
! [Q2: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_3283_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X5: nat,N2: nat] : ( modulo_modulo_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_3284_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3285_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3286_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_3287_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_3288_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% set_vebt'_def
thf(fact_3289_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ one_one_nat )
= ( M2 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_3290_finite__Collect__disjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P2 ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3291_finite__Collect__disjI,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3292_finite__Collect__disjI,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3293_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3294_finite__Collect__disjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X5: int] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P2 ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3295_finite__Collect__disjI,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3296_finite__Collect__disjI,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
& ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3297_finite__Collect__conjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P2 ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3298_finite__Collect__conjI,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3299_finite__Collect__conjI,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3300_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3301_finite__Collect__conjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P2 ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X5: int] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3302_finite__Collect__conjI,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3303_finite__Collect__conjI,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
| ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3304_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_3305_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_int @ I4 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_3306_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_3307_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_3308_dvd__0__right,axiom,
! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% dvd_0_right
thf(fact_3309_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_3310_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_3311_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_3312_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_3313_dvd__0__left__iff,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
= ( A = zero_zero_rat ) ) ).
% dvd_0_left_iff
thf(fact_3314_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_3315_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_3316_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3317_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3318_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3319_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3320_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3321_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3322_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3323_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3324_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( M2
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_3325_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_3326_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_3327_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_3328_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_3329_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_3330_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_3331_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_3332_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_3333_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_3334_div__dvd__div,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ) ).
% div_dvd_div
thf(fact_3335_div__dvd__div,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C2 )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ) ).
% div_dvd_div
thf(fact_3336_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_3337_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_3338_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_3339_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_3340_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_3341_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_3342_unset__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_3343_set__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_3344_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_3345_finite__Collect__subsets,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B7: set_nat] : ( ord_less_eq_set_nat @ B7 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3346_finite__Collect__subsets,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B7: set_complex] : ( ord_le211207098394363844omplex @ B7 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3347_finite__Collect__subsets,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite9047747110432174090at_nat
@ ( collec5514110066124741708at_nat
@ ^ [B7: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B7 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3348_finite__Collect__subsets,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B7: set_int] : ( ord_less_eq_set_int @ B7 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3349_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_3350_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_int @ I4 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_3351_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_3352_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_3353_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_3354_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_3355_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_3356_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_3357_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_3358_dvd__mult__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3359_dvd__mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3360_dvd__mult__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3361_dvd__mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3362_dvd__mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3363_dvd__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3364_dvd__mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3365_dvd__mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3366_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_3367_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_3368_dvd__add__times__triv__left__iff,axiom,
! [A: complex,C2: complex,B: complex] :
( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C2 @ A ) @ B ) )
= ( dvd_dvd_complex @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3369_dvd__add__times__triv__left__iff,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C2 @ A ) @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3370_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A ) @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3371_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3372_dvd__add__times__triv__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C2 @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3373_dvd__add__times__triv__right__iff,axiom,
! [A: complex,B: complex,C2: complex] :
( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C2 @ A ) ) )
= ( dvd_dvd_complex @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3374_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C2 @ A ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3375_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C2 @ A ) ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3376_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C2 @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3377_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C2 @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3378_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_3379_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_3380_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_3381_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_3382_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_3383_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_3384_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_3385_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_3386_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_3387_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_3388_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_3389_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_3390_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_3391_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_3392_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_3393_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_3394_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_3395_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_3396_div__add,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_3397_div__add,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_3398_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_3399_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_3400_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_3401_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_3402_mod__mult__self1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_3403_mod__mult__self1,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_3404_mod__mult__self2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_3405_mod__mult__self2,axiom,
! [A: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_3406_mod__mult__self3,axiom,
! [C2: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_3407_mod__mult__self3,axiom,
! [C2: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_3408_mod__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_3409_mod__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_3410_dvd__imp__mod__0,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( modulo_modulo_nat @ B @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_3411_dvd__imp__mod__0,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( modulo_modulo_int @ B @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_3412_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_3413_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_3414_set__encode__inverse,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_3415_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_3416_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_3417_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_3418_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_3419_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_3420_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_3421_Suc__mod__mult__self1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_3422_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_3423_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_3424_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K2 ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_3425_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_3426_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_3427_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_3428_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_3429_set__decode__0,axiom,
! [X: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_3430_zero__le__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3431_zero__le__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3432_zero__le__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3433_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_3434_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_3435_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_3436_power__less__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3437_power__less__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3438_power__less__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3439_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_3440_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_3441_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_3442_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_3443_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_3444_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_3445_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_3446_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_3447_zero__less__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3448_zero__less__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3449_zero__less__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3450_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3451_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3452_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_3453_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_3454_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3455_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3456_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3457_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3458_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_3459_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_3460_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_3461_power__le__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3462_power__le__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3463_power__le__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3464_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_3465_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_3466_int__ge__induct,axiom,
! [K2: int,I2: int,P2: int > $o] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_3467_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_3468_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_3469_zdvd__mult__cancel,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) )
=> ( ( K2 != zero_zero_int )
=> ( dvd_dvd_int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_3470_mod__0__imp__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3471_mod__0__imp__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3472_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A5: nat,B4: nat] :
( ( modulo_modulo_nat @ B4 @ A5 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3473_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A5: int,B4: int] :
( ( modulo_modulo_int @ B4 @ A5 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3474_mod__eq__0__iff__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
= ( dvd_dvd_nat @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3475_mod__eq__0__iff__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
= ( dvd_dvd_int @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3476_finite__divisors__int,axiom,
! [I2: int] :
( ( I2 != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I2 ) ) ) ) ).
% finite_divisors_int
thf(fact_3477_strict__subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( ( dvd_dvd_real @ A @ B )
& ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3478_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3479_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( ( dvd_dvd_int @ A @ B )
& ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3480_zdvd__reduce,axiom,
! [K2: int,N: int,M2: int] :
( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M2 ) ) )
= ( dvd_dvd_int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_3481_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C2: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C2 @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_3482_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_3483_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_3484_less__set__def,axiom,
( ord_le7866589430770878221at_nat
= ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
( ord_le549003669493604880_nat_o
@ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A7 )
@ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3485_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A7: set_real,B7: set_real] :
( ord_less_real_o
@ ^ [X5: real] : ( member_real @ X5 @ A7 )
@ ^ [X5: real] : ( member_real @ X5 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3486_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A7: set_set_nat,B7: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A7 )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3487_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( ord_less_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A7 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3488_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A7: set_int,B7: set_int] :
( ord_less_int_o
@ ^ [X5: int] : ( member_int @ X5 @ A7 )
@ ^ [X5: int] : ( member_int @ X5 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3489_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_3490_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_3491_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_3492_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_3493_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_3494_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X5: list_nat] : $false ) ) ).
% empty_def
thf(fact_3495_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X5: set_nat] : $false ) ) ).
% empty_def
thf(fact_3496_empty__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] : $false ) ) ).
% empty_def
thf(fact_3497_empty__def,axiom,
( bot_bot_set_real
= ( collect_real
@ ^ [X5: real] : $false ) ) ).
% empty_def
thf(fact_3498_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X5: nat] : $false ) ) ).
% empty_def
thf(fact_3499_empty__def,axiom,
( bot_bot_set_int
= ( collect_int
@ ^ [X5: int] : $false ) ) ).
% empty_def
thf(fact_3500_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3501_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3502_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_complex,R: real > complex > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3503_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3504_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3505_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_complex,R: nat > complex > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3506_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3507_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3508_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_complex,R: int > complex > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3509_pigeonhole__infinite__rel,axiom,
! [A4: set_complex,B5: set_nat,R: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A5: complex] :
( ( member_complex @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3510_not__finite__existsD,axiom,
! [P2: real > $o] :
( ~ ( finite_finite_real @ ( collect_real @ P2 ) )
=> ? [X_12: real] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3511_not__finite__existsD,axiom,
! [P2: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
=> ? [X_12: list_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3512_not__finite__existsD,axiom,
! [P2: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
=> ? [X_12: set_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3513_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_12: nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3514_not__finite__existsD,axiom,
! [P2: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P2 ) )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3515_not__finite__existsD,axiom,
! [P2: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
=> ? [X_12: complex] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3516_not__finite__existsD,axiom,
! [P2: product_prod_nat_nat > $o] :
( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
=> ? [X_12: product_prod_nat_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3517_dvd__mod__imp__dvd,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( dvd_dvd_nat @ C2 @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_3518_dvd__mod__imp__dvd,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A @ B ) )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( dvd_dvd_int @ C2 @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_3519_dvd__mod__iff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A @ B ) )
= ( dvd_dvd_nat @ C2 @ A ) ) ) ).
% dvd_mod_iff
thf(fact_3520_dvd__mod__iff,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A @ B ) )
= ( dvd_dvd_int @ C2 @ A ) ) ) ).
% dvd_mod_iff
thf(fact_3521_dvd__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_3522_dvd__trans,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_3523_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_3524_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_3525_pred__equals__eq2,axiom,
! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
( ( ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_3526_pred__equals__eq2,axiom,
! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
( ( ( ^ [X5: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_3527_pred__equals__eq2,axiom,
! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
( ( ( ^ [X5: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_3528_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_3529_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_3530_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_3531_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_3532_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_3533_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_3534_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_3535_lambda__one,axiom,
( ( ^ [X5: complex] : X5 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_3536_lambda__one,axiom,
( ( ^ [X5: real] : X5 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_3537_lambda__one,axiom,
( ( ^ [X5: rat] : X5 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_3538_lambda__one,axiom,
( ( ^ [X5: nat] : X5 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_3539_lambda__one,axiom,
( ( ^ [X5: int] : X5 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_3540_max__def__raw,axiom,
( ord_max_set_int
= ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_3541_max__def__raw,axiom,
( ord_max_rat
= ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_3542_max__def__raw,axiom,
( ord_max_num
= ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_3543_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_3544_max__def__raw,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_3545_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_3546_pred__subset__eq2,axiom,
! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
( ( ord_le5604493270027003598_nat_o
@ ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ S2 ) )
= ( ord_le3000389064537975527at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_3547_pred__subset__eq2,axiom,
! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
( ( ord_le3072208448688395470_nat_o
@ ^ [X5: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y5 ) @ S2 ) )
= ( ord_le5997549366648089703at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_3548_pred__subset__eq2,axiom,
! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
( ( ord_le3935385432712749774_nat_o
@ ^ [X5: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y5 ) @ S2 ) )
= ( ord_le1268244103169919719at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_3549_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S2 ) )
= ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_3550_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ R )
@ ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ S2 ) )
= ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_3551_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_3552_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_3553_bot__empty__eq2,axiom,
( bot_bo4898103413517107610_nat_o
= ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y5 ) @ bot_bo5327735625951526323at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_3554_bot__empty__eq2,axiom,
( bot_bo3364206721330744218_nat_o
= ( ^ [X5: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y5 ) @ bot_bo4948859079157340979at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_3555_bot__empty__eq2,axiom,
( bot_bo394778441745866138_nat_o
= ( ^ [X5: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y5 ) @ bot_bo228742789529271731at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_3556_bot__empty__eq2,axiom,
( bot_bot_int_int_o
= ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_3557_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_3558_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3559_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_3560_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_3561_mod__add__eq,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_3562_mod__add__eq,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_3563_mod__add__cong,axiom,
! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C2 )
= ( modulo_modulo_nat @ A2 @ C2 ) )
=> ( ( ( modulo_modulo_nat @ B @ C2 )
= ( modulo_modulo_nat @ B2 @ C2 ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_3564_mod__add__cong,axiom,
! [A: int,C2: int,A2: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C2 )
= ( modulo_modulo_int @ A2 @ C2 ) )
=> ( ( ( modulo_modulo_int @ B @ C2 )
= ( modulo_modulo_int @ B2 @ C2 ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_3565_mod__add__left__eq,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_3566_mod__add__left__eq,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_3567_mod__add__right__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_3568_mod__add__right__eq,axiom,
! [A: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_3569_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A5: complex,B4: complex] :
( ( A5 = zero_zero_complex )
=> ( B4 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_3570_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A5: real,B4: real] :
( ( A5 = zero_zero_real )
=> ( B4 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_3571_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A5: rat,B4: rat] :
( ( A5 = zero_zero_rat )
=> ( B4 = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_3572_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_3573_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_3574_dvd__0__left,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
=> ( A = zero_zero_rat ) ) ).
% dvd_0_left
thf(fact_3575_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_3576_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_3577_dvd__triv__right,axiom,
! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).
% dvd_triv_right
thf(fact_3578_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_3579_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_3580_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_3581_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_3582_dvd__mult__right,axiom,
! [A: complex,B: complex,C2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C2 )
=> ( dvd_dvd_complex @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_3583_dvd__mult__right,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
=> ( dvd_dvd_real @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_3584_dvd__mult__right,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C2 )
=> ( dvd_dvd_rat @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_3585_dvd__mult__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_3586_dvd__mult__right,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_3587_mult__dvd__mono,axiom,
! [A: complex,B: complex,C2: complex,D: complex] :
( ( dvd_dvd_complex @ A @ B )
=> ( ( dvd_dvd_complex @ C2 @ D )
=> ( dvd_dvd_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_3588_mult__dvd__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C2 @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_3589_mult__dvd__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ C2 @ D )
=> ( dvd_dvd_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_3590_mult__dvd__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_3591_mult__dvd__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C2 @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_3592_dvd__triv__left,axiom,
! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).
% dvd_triv_left
thf(fact_3593_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_3594_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_3595_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_3596_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_3597_dvd__mult__left,axiom,
! [A: complex,B: complex,C2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C2 )
=> ( dvd_dvd_complex @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_3598_dvd__mult__left,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
=> ( dvd_dvd_real @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_3599_dvd__mult__left,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C2 )
=> ( dvd_dvd_rat @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_3600_dvd__mult__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_3601_dvd__mult__left,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_3602_dvd__mult2,axiom,
! [A: complex,B: complex,C2: complex] :
( ( dvd_dvd_complex @ A @ B )
=> ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_3603_dvd__mult2,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_3604_dvd__mult2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_3605_dvd__mult2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_3606_dvd__mult2,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_3607_dvd__mult,axiom,
! [A: complex,C2: complex,B: complex] :
( ( dvd_dvd_complex @ A @ C2 )
=> ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_3608_dvd__mult,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ C2 )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_3609_dvd__mult,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C2 )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_3610_dvd__mult,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_3611_dvd__mult,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_3612_dvd__def,axiom,
( dvd_dvd_complex
= ( ^ [B4: complex,A5: complex] :
? [K3: complex] :
( A5
= ( times_times_complex @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_3613_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B4: real,A5: real] :
? [K3: real] :
( A5
= ( times_times_real @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_3614_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B4: rat,A5: rat] :
? [K3: rat] :
( A5
= ( times_times_rat @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_3615_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B4: nat,A5: nat] :
? [K3: nat] :
( A5
= ( times_times_nat @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_3616_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B4: int,A5: int] :
? [K3: int] :
( A5
= ( times_times_int @ B4 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_3617_dvdI,axiom,
! [A: complex,B: complex,K2: complex] :
( ( A
= ( times_times_complex @ B @ K2 ) )
=> ( dvd_dvd_complex @ B @ A ) ) ).
% dvdI
thf(fact_3618_dvdI,axiom,
! [A: real,B: real,K2: real] :
( ( A
= ( times_times_real @ B @ K2 ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_3619_dvdI,axiom,
! [A: rat,B: rat,K2: rat] :
( ( A
= ( times_times_rat @ B @ K2 ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_3620_dvdI,axiom,
! [A: nat,B: nat,K2: nat] :
( ( A
= ( times_times_nat @ B @ K2 ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_3621_dvdI,axiom,
! [A: int,B: int,K2: int] :
( ( A
= ( times_times_int @ B @ K2 ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_3622_dvdE,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ~ ! [K: complex] :
( A
!= ( times_times_complex @ B @ K ) ) ) ).
% dvdE
thf(fact_3623_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K: real] :
( A
!= ( times_times_real @ B @ K ) ) ) ).
% dvdE
thf(fact_3624_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K: rat] :
( A
!= ( times_times_rat @ B @ K ) ) ) ).
% dvdE
thf(fact_3625_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K: nat] :
( A
!= ( times_times_nat @ B @ K ) ) ) ).
% dvdE
thf(fact_3626_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K: int] :
( A
!= ( times_times_int @ B @ K ) ) ) ).
% dvdE
thf(fact_3627_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_3628_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_3629_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_3630_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_3631_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_3632_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_3633_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_3634_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_3635_one__dvd,axiom,
! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% one_dvd
thf(fact_3636_dvd__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3637_dvd__add,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3638_dvd__add,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ C2 )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3639_dvd__add,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ C2 )
=> ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3640_dvd__add__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3641_dvd__add__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3642_dvd__add__left__iff,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ C2 )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3643_dvd__add__left__iff,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C2 )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3644_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3645_dvd__add__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3646_dvd__add__right__iff,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3647_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( dvd_dvd_rat @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3648_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_3649_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_3650_dvd__div__eq__cancel,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C2 )
= ( divide_divide_nat @ B @ C2 ) )
=> ( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3651_dvd__div__eq__cancel,axiom,
! [A: int,C2: int,B: int] :
( ( ( divide_divide_int @ A @ C2 )
= ( divide_divide_int @ B @ C2 ) )
=> ( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3652_dvd__div__eq__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B @ C2 ) )
=> ( ( dvd_dvd_real @ C2 @ A )
=> ( ( dvd_dvd_real @ C2 @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_3653_dvd__div__eq__iff,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( ( divide_divide_nat @ A @ C2 )
= ( divide_divide_nat @ B @ C2 ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3654_dvd__div__eq__iff,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( ( divide_divide_int @ A @ C2 )
= ( divide_divide_int @ B @ C2 ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3655_dvd__div__eq__iff,axiom,
! [C2: real,A: real,B: real] :
( ( dvd_dvd_real @ C2 @ A )
=> ( ( dvd_dvd_real @ C2 @ B )
=> ( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B @ C2 ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_3656_mod__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_3657_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_3658_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_3659_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_3660_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_3661_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_3662_zdvd__antisym__nonneg,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M2 @ N )
=> ( ( dvd_dvd_int @ N @ M2 )
=> ( M2 = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_3663_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X5: nat] :
( collect_nat
@ ^ [N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_3664_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_3665_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_3666_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_3667_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_3668_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_3669_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).
% numeral_code(2)
thf(fact_3670_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_3671_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% set_vebt_def
thf(fact_3672_enat__0__less__mult__iff,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M2 @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M2 )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_3673_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3674_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3675_subset__decode__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% subset_decode_imp_le
thf(fact_3676_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3677_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3678_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3679_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3680_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_3681_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_3682_mod__eqE,axiom,
! [A: int,C2: int,B: int] :
( ( ( modulo_modulo_int @ A @ C2 )
= ( modulo_modulo_int @ B @ C2 ) )
=> ~ ! [D4: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C2 @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_3683_div__add1__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).
% div_add1_eq
thf(fact_3684_div__add1__eq,axiom,
! [A: int,B: int,C2: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).
% div_add1_eq
thf(fact_3685_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_3686_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_3687_minf_I10_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_3688_minf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_3689_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_3690_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_3691_minf_I9_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_3692_minf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_3693_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_3694_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% minf(9)
thf(fact_3695_pinf_I10_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_3696_pinf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_3697_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_3698_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_3699_pinf_I9_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_3700_pinf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z3 @ X3 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_3701_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_3702_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% pinf(9)
thf(fact_3703_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3704_dvd__div__eq__0__iff,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3705_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3706_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3707_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3708_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_3709_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_3710_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff
thf(fact_3711_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff
thf(fact_3712_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% mult_unit_dvd_iff
thf(fact_3713_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% mult_unit_dvd_iff
thf(fact_3714_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_3715_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_3716_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_3717_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_3718_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% unit_mult_left_cancel
thf(fact_3719_unit__mult__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% unit_mult_left_cancel
thf(fact_3720_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_mult_right_cancel
thf(fact_3721_unit__mult__right__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_mult_right_cancel
thf(fact_3722_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_3723_finite__set__decode,axiom,
! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_3724_mod__induct,axiom,
! [P2: nat > $o,N: nat,P: nat,M2: nat] :
( ( P2 @ N )
=> ( ( ord_less_nat @ N @ P )
=> ( ( ord_less_nat @ M2 @ P )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P )
=> ( ( P2 @ N3 )
=> ( P2 @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P ) ) ) )
=> ( P2 @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_3725_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ D @ C2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C2 @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_3726_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C2: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ D @ C2 )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C2 @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_3727_dvd__mult__imp__div,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ B )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) ) ).
% dvd_mult_imp_div
thf(fact_3728_dvd__mult__imp__div,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ B )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C2 ) ) ) ).
% dvd_mult_imp_div
thf(fact_3729_dvd__div__mult2__eq,axiom,
! [B: nat,C2: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C2 ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).
% dvd_div_mult2_eq
thf(fact_3730_dvd__div__mult2__eq,axiom,
! [B: int,C2: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C2 ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% dvd_div_mult2_eq
thf(fact_3731_div__div__eq__right,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).
% div_div_eq_right
thf(fact_3732_div__div__eq__right,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C2 ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).
% div_div_eq_right
thf(fact_3733_div__mult__swap,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ) ).
% div_mult_swap
thf(fact_3734_div__mult__swap,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 ) ) ) ).
% div_mult_swap
thf(fact_3735_dvd__div__mult,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C2 ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C2 ) ) ) ).
% dvd_div_mult
thf(fact_3736_dvd__div__mult,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C2 ) @ A )
= ( divide_divide_int @ ( times_times_int @ B @ A ) @ C2 ) ) ) ).
% dvd_div_mult
thf(fact_3737_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_3738_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [M: nat] : ( P2 @ M @ zero_zero_nat )
=> ( ! [M: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 @ ( modulo_modulo_nat @ M @ N3 ) )
=> ( P2 @ M @ N3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_3739_unit__div__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_div_cancel
thf(fact_3740_unit__div__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% unit_div_cancel
thf(fact_3741_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% div_unit_dvd_iff
thf(fact_3742_div__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% div_unit_dvd_iff
thf(fact_3743_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C2 @ B ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_div_unit_iff
thf(fact_3744_dvd__div__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_div_unit_iff
thf(fact_3745_div__plus__div__distrib__dvd__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3746_div__plus__div__distrib__dvd__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3747_div__plus__div__distrib__dvd__right,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3748_div__plus__div__distrib__dvd__right,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3749_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_3750_mod__eq__0D,axiom,
! [M2: nat,D: nat] :
( ( ( modulo_modulo_nat @ M2 @ D )
= zero_zero_nat )
=> ? [Q3: nat] :
( M2
= ( times_times_nat @ D @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_3751_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M2: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3752_dvd__power__le,axiom,
! [X: real,Y: real,N: nat,M2: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3753_dvd__power__le,axiom,
! [X: complex,Y: complex,N: nat,M2: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3754_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M2: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3755_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3756_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M2: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3757_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M2: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3758_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M2: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3759_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3760_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3761_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3762_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3763_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_3764_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_3765_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_3766_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_3767_zdvd__not__zless,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_int @ M2 @ N )
=> ~ ( dvd_dvd_int @ N @ M2 ) ) ) ).
% zdvd_not_zless
thf(fact_3768_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_3769_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) )
| ( ( times_times_nat @ B @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_3770_q__pos__lemma,axiom,
! [B2: int,Q6: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_3771_zdiv__mono2__lemma,axiom,
! [B: int,Q2: int,R2: int,B2: int,Q6: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B2 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_3772_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q2: int,R2: int,B2: int,Q6: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B2 @ Q6 ) @ R4 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q6 ) @ R4 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_3773_unique__quotient__lemma,axiom,
! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_3774_unique__quotient__lemma__neg,axiom,
! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R4 )
=> ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_3775_incr__mult__lemma,axiom,
! [D: int,P2: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( plus_plus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_3776_zmult__zless__mono2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I2 ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_3777_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_3778_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_3779_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_3780_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_3781_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_3782_finite__lists__length__eq,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs3: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A4 )
& ( ( size_s3451745648224563538omplex @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3783_finite__lists__length__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat,N: nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite500796754983035824at_nat
@ ( collec3343600615725829874at_nat
@ ^ [Xs3: list_P6011104703257516679at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A4 )
& ( ( size_s5460976970255530739at_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3784_finite__lists__length__eq,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs3: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A4 )
& ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3785_finite__lists__length__eq,axiom,
! [A4: set_o,N: nat] :
( ( finite_finite_o @ A4 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs3: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs3 ) @ A4 )
& ( ( size_size_list_o @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3786_finite__lists__length__eq,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3787_finite__lists__length__eq,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A4 )
& ( ( size_size_list_int @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3788_even__set__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( M2 != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_3789_even__set__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( M2 != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_3790_even__unset__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M2 = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_3791_even__unset__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M2 = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_3792_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_3793_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( modulo_modulo_nat @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3794_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( modulo_modulo_int @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3795_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3796_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3797_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3798_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3799_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_3800_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_3801_finite__lists__length__le,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs3: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3802_finite__lists__length__le,axiom,
! [A4: set_Pr1261947904930325089at_nat,N: nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite500796754983035824at_nat
@ ( collec3343600615725829874at_nat
@ ^ [Xs3: list_P6011104703257516679at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3803_finite__lists__length__le,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs3: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3804_finite__lists__length__le,axiom,
! [A4: set_o,N: nat] :
( ( finite_finite_o @ A4 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs3: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3805_finite__lists__length__le,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3806_finite__lists__length__le,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3807_div__mult1__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).
% div_mult1_eq
thf(fact_3808_div__mult1__eq,axiom,
! [A: int,B: int,C2: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).
% div_mult1_eq
thf(fact_3809_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3810_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3811_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3812_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3813_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3814_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3815_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3816_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3817_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3818_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3819_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
= ( plus_plus_nat @ A @ C2 ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3820_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
= ( plus_plus_int @ A @ C2 ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3821_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
= ( plus_plus_nat @ A @ C2 ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3822_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
= ( plus_plus_int @ A @ C2 ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3823_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C: nat] :
( B
!= ( times_times_nat @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_3824_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C: int] :
( B
!= ( times_times_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_3825_unity__coeff__ex,axiom,
! [P2: complex > $o,L: complex] :
( ( ? [X5: complex] : ( P2 @ ( times_times_complex @ L @ X5 ) ) )
= ( ? [X5: complex] :
( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X5 @ zero_zero_complex ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3826_unity__coeff__ex,axiom,
! [P2: real > $o,L: real] :
( ( ? [X5: real] : ( P2 @ ( times_times_real @ L @ X5 ) ) )
= ( ? [X5: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X5 @ zero_zero_real ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3827_unity__coeff__ex,axiom,
! [P2: rat > $o,L: rat] :
( ( ? [X5: rat] : ( P2 @ ( times_times_rat @ L @ X5 ) ) )
= ( ? [X5: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X5 @ zero_zero_rat ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3828_unity__coeff__ex,axiom,
! [P2: nat > $o,L: nat] :
( ( ? [X5: nat] : ( P2 @ ( times_times_nat @ L @ X5 ) ) )
= ( ? [X5: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X5 @ zero_zero_nat ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3829_unity__coeff__ex,axiom,
! [P2: int > $o,L: int] :
( ( ? [X5: int] : ( P2 @ ( times_times_int @ L @ X5 ) ) )
= ( ? [X5: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X5 @ zero_zero_int ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3830_dvd__div__div__eq__mult,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D @ C2 ) )
= ( ( times_times_nat @ B @ C2 )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_3831_dvd__div__div__eq__mult,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C2 != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C2 @ D )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D @ C2 ) )
= ( ( times_times_int @ B @ C2 )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_3832_dvd__div__iff__mult,axiom,
! [C2: nat,B: nat,A: nat] :
( ( C2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_3833_dvd__div__iff__mult,axiom,
! [C2: int,B: int,A: int] :
( ( C2 != zero_zero_int )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C2 ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_3834_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_3835_div__dvd__iff__mult,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_3836_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C2 )
= ( B
= ( times_times_nat @ C2 @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_3837_dvd__div__eq__mult,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C2 )
= ( B
= ( times_times_int @ C2 @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_3838_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3839_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3840_unit__eq__div1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= C2 )
= ( A
= ( times_times_nat @ C2 @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_3841_unit__eq__div1,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= C2 )
= ( A
= ( times_times_int @ C2 @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_3842_unit__eq__div2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( A
= ( divide_divide_nat @ C2 @ B ) )
= ( ( times_times_nat @ A @ B )
= C2 ) ) ) ).
% unit_eq_div2
thf(fact_3843_unit__eq__div2,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( A
= ( divide_divide_int @ C2 @ B ) )
= ( ( times_times_int @ A @ B )
= C2 ) ) ) ).
% unit_eq_div2
thf(fact_3844_div__mult__unit2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).
% div_mult_unit2
thf(fact_3845_div__mult__unit2,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).
% div_mult_unit2
thf(fact_3846_unit__div__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
= ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ) ).
% unit_div_commute
thf(fact_3847_unit__div__commute,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C2 )
= ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ B ) ) ) ).
% unit_div_commute
thf(fact_3848_unit__div__mult__swap,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ) ).
% unit_div_mult_swap
thf(fact_3849_unit__div__mult__swap,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 ) ) ) ).
% unit_div_mult_swap
thf(fact_3850_is__unit__div__mult2__eq,axiom,
! [B: nat,C2: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ C2 @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_3851_is__unit__div__mult2__eq,axiom,
! [B: int,C2: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ C2 @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_3852_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_3853_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3854_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3855_div__less__mono,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A4 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_3856_mod__eq__nat1E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ~ ! [S3: nat] :
( M2
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_3857_mod__eq__nat2E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus_nat @ M2 @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_3858_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_3859_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_3860_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N @ Q2 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_3861_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_3862_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_3863_dvd__mult__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_3864_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_3865_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_3866_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_3867_split__zdiv,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( divide_divide_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( P2 @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_3868_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_3869_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_3870_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_3871_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3872_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3873_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3874_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3875_is__unitE,axiom,
! [A: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B3 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B3 )
= A )
=> ( ( ( times_times_nat @ A @ B3 )
= one_one_nat )
=> ( ( divide_divide_nat @ C2 @ A )
!= ( times_times_nat @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3876_is__unitE,axiom,
! [A: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B3: int] :
( ( B3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B3 )
=> ( ( ( divide_divide_int @ one_one_int @ B3 )
= A )
=> ( ( ( times_times_int @ A @ B3 )
= one_one_int )
=> ( ( divide_divide_int @ C2 @ A )
!= ( times_times_int @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3877_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_3878_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_3879_dvd__power__iff,axiom,
! [X: nat,M2: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_3880_dvd__power__iff,axiom,
! [X: int,M2: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_3881_dvd__power,axiom,
! [N: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% dvd_power
thf(fact_3882_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_3883_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_3884_dvd__power,axiom,
! [N: nat,X: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_complex ) )
=> ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% dvd_power
thf(fact_3885_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_3886_split__mod,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ M2 ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
=> ( P2 @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_3887_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_3888_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_3889_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_3890_power__dvd__imp__le,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_3891_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3892_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3893_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3894_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3895_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3896_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3897_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3898_power__mono__odd,axiom,
! [N: nat,A: rat,B: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3899_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3900_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_3901_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_3902_dvd__power__iff__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_3903_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3904_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3905_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_3906_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_3907_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_3908_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M2: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3909_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M2: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3910_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_3911_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_3912_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3913_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3914_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3915_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3916_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3917_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3918_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3919_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3920_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3921_verit__le__mono__div,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_3922_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_3923_even__set__encode__iff,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3924_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3925_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3926_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less_nat @ X @ Mi )
& ( ~ ( ord_less_nat @ X @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ X )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_3927_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_3928_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3929_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3930_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3931_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_3932_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_3933_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_3934_mod__double__modulus,axiom,
! [M2: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_nat @ X @ M2 ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3935_mod__double__modulus,axiom,
! [M2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_int @ X @ M2 ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3936_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_3937_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3938_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3939_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3940_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_3941_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_3942_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_3943_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_3944_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_3945_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_3946_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
= ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3947_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
= ( divide_divide_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3948_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3949_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3950_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3951_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_3952_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z5: real] :
( ( power_power_real @ Z5 @ N )
= one_one_real ) ) ) ) ).
% finite_roots_unity
thf(fact_3953_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) ) ) ) ).
% finite_roots_unity
thf(fact_3954_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_3955_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3956_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3957_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3958_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3959_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3960_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3961_product__nth,axiom,
! [N: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N )
= ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3962_product__nth,axiom,
! [N: nat,Xs: list_o,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N )
= ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3963_product__nth,axiom,
! [N: nat,Xs: list_o,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N )
= ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3964_product__nth,axiom,
! [N: nat,Xs: list_o,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N )
= ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3965_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N )
= ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3966_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N )
= ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3967_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > complex,Y: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3968_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3969_prod_Ofinite__Collect__op,axiom,
! [I6: set_int,X: int > complex,Y: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3970_prod_Ofinite__Collect__op,axiom,
! [I6: set_complex,X: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3971_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > real,Y: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3972_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3973_prod_Ofinite__Collect__op,axiom,
! [I6: set_int,X: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3974_prod_Ofinite__Collect__op,axiom,
! [I6: set_complex,X: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3975_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > rat,Y: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3976_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= one_one_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3977_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > complex,Y: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3978_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3979_sum_Ofinite__Collect__op,axiom,
! [I6: set_int,X: int > complex,Y: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3980_sum_Ofinite__Collect__op,axiom,
! [I6: set_complex,X: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3981_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > real,Y: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3982_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3983_sum_Ofinite__Collect__op,axiom,
! [I6: set_int,X: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I6 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3984_sum_Ofinite__Collect__op,axiom,
! [I6: set_complex,X: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I6 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3985_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X: real > rat,Y: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I6 )
& ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3986_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( X @ I4 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( Y @ I4 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I6 )
& ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3987_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3988_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3989_flip__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_3990_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_3991_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_3992_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3993_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_o] :
( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3994_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_nat] :
( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3995_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_int] :
( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_3996_length__product,axiom,
! [Xs: list_o,Ys: list_VEBT_VEBT] :
( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3997_length__product,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3998_length__product,axiom,
! [Xs: list_o,Ys: list_nat] :
( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3999_length__product,axiom,
! [Xs: list_o,Ys: list_int] :
( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_4000_length__product,axiom,
! [Xs: list_nat,Ys: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_4001_length__product,axiom,
! [Xs: list_nat,Ys: list_o] :
( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_4002_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_4003_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_4004_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_4005_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_4006_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_4007_zmod__trivial__iff,axiom,
! [I2: int,K2: int] :
( ( ( modulo_modulo_int @ I2 @ K2 )
= I2 )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K2 ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K2 @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_4008_pos__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
& ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% pos_mod_conj
thf(fact_4009_neg__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% neg_mod_conj
thf(fact_4010_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K2 ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_4011_zdiv__mono__strict,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A4 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_4012_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( plus_plus_int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_4013_int__mod__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_4014_int__mod__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_4015_split__zmod,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( modulo_modulo_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( P2 @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_4016_split__neg__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_4017_split__pos__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_4018_verit__le__mono__div__int,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_4019_even__flip__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M2 = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4020_even__flip__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M2 = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4021_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_4022_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_4023_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_4024_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_4025_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_4026_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_4027_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_4028_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_4029_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_4030_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_4031_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_4032_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4033_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4034_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4035_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M2 )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4036_even__mult__exp__div__exp__iff,axiom,
! [A: int,M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M2 )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4037_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_4038_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4039_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4040_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4041_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4042_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4043_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_4044_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_4045_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_4046_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_4047_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_4048_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_4049_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_4050_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_4051_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_4052_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_4053_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_4054_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_4055_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_4056_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_4057_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4058_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4059_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4060_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4061_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4062_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4063_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4064_add__diff__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4065_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4066_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4067_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4068_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4069_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4070_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4071_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4072_add__diff__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4073_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_4074_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_4075_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_4076_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_4077_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_4078_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_4079_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_4080_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_4081_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_4082_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_4083_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_4084_diff__diff__left,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_4085_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4086_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4087_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4088_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n1201886186963655149omplex @ P2 )
= zero_zero_complex )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_4089_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n3304061248610475627l_real @ P2 )
= zero_zero_real )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_4090_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P2 )
= zero_zero_rat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_4091_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= zero_zero_nat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_4092_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= zero_zero_int )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_4093_of__bool__eq_I1_J,axiom,
( ( zero_n1201886186963655149omplex @ $false )
= zero_zero_complex ) ).
% of_bool_eq(1)
thf(fact_4094_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_4095_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_4096_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_4097_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_4098_of__bool__less__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) )
= ( ~ P2
& Q ) ) ).
% of_bool_less_iff
thf(fact_4099_of__bool__less__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( ~ P2
& Q ) ) ).
% of_bool_less_iff
thf(fact_4100_of__bool__less__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( ~ P2
& Q ) ) ).
% of_bool_less_iff
thf(fact_4101_of__bool__less__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( ~ P2
& Q ) ) ).
% of_bool_less_iff
thf(fact_4102_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_4103_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_4104_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_4105_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_4106_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_4107_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n3304061248610475627l_real @ P2 )
= one_one_real )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_4108_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n1201886186963655149omplex @ P2 )
= one_one_complex )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_4109_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P2 )
= one_one_rat )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_4110_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= one_one_nat )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_4111_of__bool__eq__1__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= one_one_int )
= P2 ) ).
% of_bool_eq_1_iff
thf(fact_4112_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_4113_of__bool__or__iff,axiom,
! [P2: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P2
| Q ) )
= ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_or_iff
thf(fact_4114_of__bool__or__iff,axiom,
! [P2: $o,Q: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P2
| Q ) )
= ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% of_bool_or_iff
thf(fact_4115_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_4116_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_4117_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_4118_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_4119_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_rat @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_4120_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_4121_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_4122_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_4123_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_4124_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_4125_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_4126_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_4127_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_4128_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_4129_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_4130_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_4131_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_4132_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_4133_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_4134_div__diff,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).
% div_diff
thf(fact_4135_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4136_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4137_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4138_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4139_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_4140_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_4141_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_4142_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_4143_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_4144_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_4145_of__bool__less__one__iff,axiom,
! [P2: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
= ~ P2 ) ).
% of_bool_less_one_iff
thf(fact_4146_of__bool__not__iff,axiom,
! [P2: $o] :
( ( zero_n1201886186963655149omplex @ ~ P2 )
= ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P2 ) ) ) ).
% of_bool_not_iff
thf(fact_4147_of__bool__not__iff,axiom,
! [P2: $o] :
( ( zero_n3304061248610475627l_real @ ~ P2 )
= ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ) ).
% of_bool_not_iff
thf(fact_4148_of__bool__not__iff,axiom,
! [P2: $o] :
( ( zero_n2052037380579107095ol_rat @ ~ P2 )
= ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ) ).
% of_bool_not_iff
thf(fact_4149_of__bool__not__iff,axiom,
! [P2: $o] :
( ( zero_n2684676970156552555ol_int @ ~ P2 )
= ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ) ).
% of_bool_not_iff
thf(fact_4150_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_4151_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_4152_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_4153_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_4154_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_4155_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_4156_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_4157_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_4158_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_4159_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_4160_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_4161_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_4162_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_4163_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_4164_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4165_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4166_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_4167_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_4168_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_4169_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_4170_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4171_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4172_of__bool__eq__iff,axiom,
! [P: $o,Q2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( P = Q2 ) ) ).
% of_bool_eq_iff
thf(fact_4173_of__bool__eq__iff,axiom,
! [P: $o,Q2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= ( zero_n2684676970156552555ol_int @ Q2 ) )
= ( P = Q2 ) ) ).
% of_bool_eq_iff
thf(fact_4174_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4175_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4176_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4177_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4178_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4179_diff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4180_diff__right__commute,axiom,
! [A: rat,C2: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4181_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_4182_of__bool__conj,axiom,
! [P2: $o,Q: $o] :
( ( zero_n1201886186963655149omplex
@ ( P2
& Q ) )
= ( times_times_complex @ ( zero_n1201886186963655149omplex @ P2 ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).
% of_bool_conj
thf(fact_4183_of__bool__conj,axiom,
! [P2: $o,Q: $o] :
( ( zero_n3304061248610475627l_real
@ ( P2
& Q ) )
= ( times_times_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% of_bool_conj
thf(fact_4184_of__bool__conj,axiom,
! [P2: $o,Q: $o] :
( ( zero_n2052037380579107095ol_rat
@ ( P2
& Q ) )
= ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% of_bool_conj
thf(fact_4185_of__bool__conj,axiom,
! [P2: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P2
& Q ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_conj
thf(fact_4186_of__bool__conj,axiom,
! [P2: $o,Q: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P2
& Q ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% of_bool_conj
thf(fact_4187_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4188_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( ord_less_eq_rat @ A @ B )
= ( ord_less_eq_rat @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4189_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4190_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4191_diff__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4192_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4193_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4194_diff__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4195_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4196_diff__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4197_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ D @ C2 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4198_diff__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4199_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: complex,Z2: complex] : Y4 = Z2 )
= ( ^ [A5: complex,B4: complex] :
( ( minus_minus_complex @ A5 @ B4 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4200_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4201_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
= ( ^ [A5: real,B4: real] :
( ( minus_minus_real @ A5 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4202_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
= ( ^ [A5: rat,B4: rat] :
( ( minus_minus_rat @ A5 @ B4 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4203_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4204_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4205_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4206_diff__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4207_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4208_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4209_diff__eq__diff__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4210_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( ord_less_rat @ A @ B )
= ( ord_less_rat @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4211_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4212_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4213_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ D @ C2 )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4214_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4215_left__diff__distrib,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C2 )
= ( minus_minus_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_4216_left__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_4217_left__diff__distrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_4218_left__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_4219_right__diff__distrib,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_4220_right__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_4221_right__diff__distrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_4222_right__diff__distrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_4223_left__diff__distrib_H,axiom,
! [B: complex,C2: complex,A: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B @ C2 ) @ A )
= ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_4224_left__diff__distrib_H,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_4225_left__diff__distrib_H,axiom,
! [B: rat,C2: rat,A: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B @ C2 ) @ A )
= ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_4226_left__diff__distrib_H,axiom,
! [B: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_4227_left__diff__distrib_H,axiom,
! [B: int,C2: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C2 ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_4228_right__diff__distrib_H,axiom,
! [A: complex,B: complex,C2: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_4229_right__diff__distrib_H,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_4230_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_4231_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_4232_right__diff__distrib_H,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_4233_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4234_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4235_diff__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4236_diff__diff__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4237_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4238_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4239_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4240_add__implies__diff,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ( plus_plus_rat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_rat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4241_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4242_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4243_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4244_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4245_diff__add__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4246_diff__add__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4247_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4248_diff__diff__eq2,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4249_diff__diff__eq2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4250_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4251_add__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4252_add__diff__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4253_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4254_eq__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4255_eq__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( A
= ( minus_minus_rat @ C2 @ B ) )
= ( ( plus_plus_rat @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4256_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4257_diff__eq__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4258_diff__eq__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( minus_minus_rat @ A @ B )
= C2 )
= ( A
= ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4259_group__cancel_Osub1,axiom,
! [A4: int,K2: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A4 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4260_group__cancel_Osub1,axiom,
! [A4: real,K2: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( minus_minus_real @ A4 @ B )
= ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4261_group__cancel_Osub1,axiom,
! [A4: rat,K2: rat,A: rat,B: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( minus_minus_rat @ A4 @ B )
= ( plus_plus_rat @ K2 @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4262_add__diff__add,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4263_add__diff__add,axiom,
! [A: real,C2: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4264_add__diff__add,axiom,
! [A: rat,C2: rat,B: rat,D: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4265_diff__divide__distrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ).
% diff_divide_distrib
thf(fact_4266_diff__divide__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% diff_divide_distrib
thf(fact_4267_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_4268_dvd__diff,axiom,
! [X: real,Y: real,Z: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_4269_dvd__diff,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( ( dvd_dvd_rat @ X @ Z )
=> ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_4270_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_4271_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_4272_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_4273_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_4274_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_4275_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_4276_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_4277_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_4278_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_4279_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_4280_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_4281_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_4282_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_4283_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_4284_diff__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_4285_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_4286_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_4287_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_4288_max__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_4289_max__diff__distrib__left,axiom,
! [X: real,Y: real,Z: real] :
( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_4290_max__diff__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_4291_dvd__diff__nat,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_4292_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4293_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4294_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4295_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4296_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_4297_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat ) ).
% of_bool_less_eq_one
thf(fact_4298_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_4299_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_4300_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_4301_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_4302_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_4303_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_4304_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_4305_split__of__bool,axiom,
! [P2: complex > $o,P: $o] :
( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
= ( ( P
=> ( P2 @ one_one_complex ) )
& ( ~ P
=> ( P2 @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_4306_split__of__bool,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ( P
=> ( P2 @ one_one_real ) )
& ( ~ P
=> ( P2 @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_4307_split__of__bool,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ( P
=> ( P2 @ one_one_rat ) )
& ( ~ P
=> ( P2 @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_4308_split__of__bool,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ( P
=> ( P2 @ one_one_nat ) )
& ( ~ P
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_4309_split__of__bool,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ( P
=> ( P2 @ one_one_int ) )
& ( ~ P
=> ( P2 @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_4310_split__of__bool__asm,axiom,
! [P2: complex > $o,P: $o] :
( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_complex ) )
| ( ~ P
& ~ ( P2 @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4311_split__of__bool__asm,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_real ) )
| ( ~ P
& ~ ( P2 @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4312_split__of__bool__asm,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_rat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4313_split__of__bool__asm,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_nat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4314_split__of__bool__asm,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_int ) )
| ( ~ P
& ~ ( P2 @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4315_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_4316_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_4317_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_4318_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_4319_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_4320_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_4321_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4322_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_4323_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4324_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4325_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4326_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4327_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4328_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4329_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_4330_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_4331_le__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4332_le__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4333_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4334_diff__le__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4335_diff__le__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4336_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4337_add__le__add__imp__diff__le,axiom,
! [I2: real,K2: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4338_add__le__add__imp__diff__le,axiom,
! [I2: rat,K2: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4339_add__le__add__imp__diff__le,axiom,
! [I2: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4340_add__le__add__imp__diff__le,axiom,
! [I2: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4341_add__le__imp__le__diff,axiom,
! [I2: real,K2: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4342_add__le__imp__le__diff,axiom,
! [I2: rat,K2: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
=> ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4343_add__le__imp__le__diff,axiom,
! [I2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4344_add__le__imp__le__diff,axiom,
! [I2: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4345_less__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4346_less__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4347_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4348_diff__less__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4349_diff__less__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( ord_less_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4350_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4351_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4352_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ ( ord_less_rat @ A @ B )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4353_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4354_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4355_square__diff__square__factored,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
= ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4356_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4357_square__diff__square__factored,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4358_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4359_eq__add__iff2,axiom,
! [A: complex,E2: complex,C2: complex,B: complex,D: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4360_eq__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4361_eq__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4362_eq__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4363_eq__add__iff1,axiom,
! [A: complex,E2: complex,C2: complex,B: complex,D: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
= ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4364_eq__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4365_eq__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4366_eq__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4367_mult__diff__mult,axiom,
! [X: complex,Y: complex,A: complex,B: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4368_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4369_mult__diff__mult,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4370_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4371_dvd__minus__mod,axiom,
! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_4372_dvd__minus__mod,axiom,
! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_4373_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_4374_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_4375_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_4376_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_4377_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_4378_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_4379_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_4380_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_4381_less__diff__conv,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_4382_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_4383_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_4384_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_4385_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_4386_le__diff__conv,axiom,
! [J: nat,K2: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_4387_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_4388_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
= ( ( ord_less_nat @ N @ M2 )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_4389_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
= ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_4390_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_4391_dvd__diffD,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_4392_mod__geq,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ M2 @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_4393_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_4394_le__mod__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_4395_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
= ( ord_max_nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_4396_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4397_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4398_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4399_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4400_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4401_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4402_less__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4403_less__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4404_less__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4405_less__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4406_less__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4407_less__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4408_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4409_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4410_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4411_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4412_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4413_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4414_diff__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_4415_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_4416_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_4417_divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_4418_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_4419_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_4420_square__diff__one__factored,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_4421_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_4422_square__diff__one__factored,axiom,
! [X: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_4423_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_4424_inf__period_I3_J,axiom,
! [D: complex,D6: complex,T: complex] :
( ( dvd_dvd_complex @ D @ D6 )
=> ! [X3: complex,K4: complex] :
( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X3 @ T ) )
= ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4425_inf__period_I3_J,axiom,
! [D: real,D6: real,T: real] :
( ( dvd_dvd_real @ D @ D6 )
=> ! [X3: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4426_inf__period_I3_J,axiom,
! [D: rat,D6: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D6 )
=> ! [X3: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4427_inf__period_I3_J,axiom,
! [D: int,D6: int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X3: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4428_inf__period_I4_J,axiom,
! [D: complex,D6: complex,T: complex] :
( ( dvd_dvd_complex @ D @ D6 )
=> ! [X3: complex,K4: complex] :
( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4429_inf__period_I4_J,axiom,
! [D: real,D6: real,T: real] :
( ( dvd_dvd_real @ D @ D6 )
=> ! [X3: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4430_inf__period_I4_J,axiom,
! [D: rat,D6: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D6 )
=> ! [X3: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4431_inf__period_I4_J,axiom,
! [D: int,D6: int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X3: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4432_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_4433_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_4434_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_4435_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_4436_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_4437_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_4438_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_4439_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_4440_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_4441_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P2 @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_4442_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P2 @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_4443_less__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_4444_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_4445_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_4446_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_4447_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_4448_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_4449_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_4450_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( ( modulo_modulo_nat @ M2 @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_4451_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M2 ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_4452_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M2 ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_4453_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_4454_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_4455_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_4456_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_4457_signed__take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_4458_power__diff,axiom,
! [A: rat,N: nat,M2: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4459_power__diff,axiom,
! [A: complex,N: nat,M2: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4460_power__diff,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4461_power__diff,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4462_power__diff,axiom,
! [A: real,N: nat,M2: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4463_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_4464_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_4465_div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_4466_div__if,axiom,
( divide_divide_nat
= ( ^ [M3: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M3 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_4467_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_4468_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_4469_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_4470_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_4471_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R2: nat,M2: nat] :
( ( ord_less_eq_nat @ Q2 @ N )
=> ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M2 ) )
=> ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_4472_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M2 )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
=> ( ( modulo_modulo_nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_4473_scaling__mono,axiom,
! [U: real,V: real,R2: real,S: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_4474_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_4475_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4476_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4477_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_4478_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_4479_power__diff__power__eq,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4480_power__diff__power__eq,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4481_power__eq__if,axiom,
( power_power_complex
= ( ^ [P6: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4482_power__eq__if,axiom,
( power_power_real
= ( ^ [P6: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4483_power__eq__if,axiom,
( power_power_rat
= ( ^ [P6: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4484_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4485_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4486_power__minus__mult,axiom,
! [N: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4487_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4488_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4489_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4490_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4491_diff__le__diff__pow,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_4492_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_4493_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_4494_bits__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [A3: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ A3 ) )
=> ( ! [A3: nat,B3: $o] :
( ( P2 @ A3 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P2 @ A ) ) ) ).
% bits_induct
thf(fact_4495_bits__induct,axiom,
! [P2: int > $o,A: int] :
( ! [A3: int] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ A3 ) )
=> ( ! [A3: int,B3: $o] :
( ( P2 @ A3 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P2 @ A ) ) ) ).
% bits_induct
thf(fact_4496_int__power__div__base,axiom,
! [M2: nat,K2: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( divide_divide_int @ ( power_power_int @ K2 @ M2 ) @ K2 )
= ( power_power_int @ K2 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_4497_exp__mod__exp,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% exp_mod_exp
thf(fact_4498_exp__mod__exp,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% exp_mod_exp
thf(fact_4499_power2__diff,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_4500_power2__diff,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_4501_power2__diff,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_4502_power2__diff,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_4503_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_4504_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_4505_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_4506_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_4507_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% even_mask_div_iff'
thf(fact_4508_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% even_mask_div_iff'
thf(fact_4509_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_4510_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4511_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4512_divmod__step__eq,axiom,
! [L: num,R2: int,Q2: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_4513_divmod__step__eq,axiom,
! [L: num,R2: nat,Q2: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_4514_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% inrange
thf(fact_4515_artanh__def,axiom,
( artanh_real
= ( ^ [X5: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X5 ) @ ( minus_minus_real @ one_one_real @ X5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_4516_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,A5: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4517_diff__shunt__var,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ X @ Y )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_4518_diff__shunt__var,axiom,
! [X: set_real,Y: set_real] :
( ( ( minus_minus_set_real @ X @ Y )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_4519_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_4520_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_4521_take__bit__rec,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N2: nat,A5: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_4522_take__bit__rec,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N2: nat,A5: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_4523_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_4524_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4525_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4526_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4527_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4528_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4529_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4530_Diff__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A4 @ bot_bo2099793752762293965at_nat )
= A4 ) ).
% Diff_empty
thf(fact_4531_Diff__empty,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ A4 @ bot_bot_set_real )
= A4 ) ).
% Diff_empty
thf(fact_4532_Diff__empty,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
= A4 ) ).
% Diff_empty
thf(fact_4533_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_4534_empty__Diff,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A4 )
= bot_bo2099793752762293965at_nat ) ).
% empty_Diff
thf(fact_4535_empty__Diff,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A4 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_4536_empty__Diff,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_4537_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_4538_Diff__cancel,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A4 @ A4 )
= bot_bo2099793752762293965at_nat ) ).
% Diff_cancel
thf(fact_4539_Diff__cancel,axiom,
! [A4: set_real] :
( ( minus_minus_set_real @ A4 @ A4 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_4540_Diff__cancel,axiom,
! [A4: set_int] :
( ( minus_minus_set_int @ A4 @ A4 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_4541_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_4542_finite__Diff,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4543_finite__Diff,axiom,
! [A4: set_complex,B5: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4544_finite__Diff,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4545_finite__Diff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4546_finite__Diff2,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( finite_finite_int @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4547_finite__Diff2,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4548_finite__Diff2,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
= ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4549_finite__Diff2,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4550_compl__le__compl__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
= ( ord_less_eq_set_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_4551_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4552_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4553_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4554_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_4555_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_4556_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_4557_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_4558_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4559_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4560_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4561_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4562_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_4563_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_4564_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_4565_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_4566_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_4567_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_4568_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_4569_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_4570_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_4571_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_4572_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4573_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4574_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4575_mult__minus__right,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4576_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4577_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4578_mult__minus__right,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4579_minus__mult__minus,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( times_times_complex @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4580_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4581_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4582_minus__mult__minus,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( times_times_rat @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4583_mult__minus__left,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4584_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4585_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4586_mult__minus__left,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4587_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4588_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4589_add__minus__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4590_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4591_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4592_minus__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4593_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_4594_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_4595_minus__add__distrib,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_4596_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4597_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4598_minus__diff__eq,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
= ( minus_minus_rat @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4599_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( dvd_dvd_int @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_4600_minus__dvd__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
= ( dvd_dvd_real @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_4601_minus__dvd__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_4602_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_4603_dvd__minus__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
= ( dvd_dvd_real @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_4604_dvd__minus__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_4605_Diff__eq__empty__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4606_Diff__eq__empty__iff,axiom,
! [A4: set_real,B5: set_real] :
( ( ( minus_minus_set_real @ A4 @ B5 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4607_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4608_Diff__eq__empty__iff,axiom,
! [A4: set_int,B5: set_int] :
( ( ( minus_minus_set_int @ A4 @ B5 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4609_Icc__eq__Icc,axiom,
! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
( ( ( set_or370866239135849197et_int @ L @ H2 )
= ( set_or370866239135849197et_int @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_int @ L @ H2 )
& ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4610_Icc__eq__Icc,axiom,
! [L: rat,H2: rat,L3: rat,H3: rat] :
( ( ( set_or633870826150836451st_rat @ L @ H2 )
= ( set_or633870826150836451st_rat @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_rat @ L @ H2 )
& ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4611_Icc__eq__Icc,axiom,
! [L: num,H2: num,L3: num,H3: num] :
( ( ( set_or7049704709247886629st_num @ L @ H2 )
= ( set_or7049704709247886629st_num @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_num @ L @ H2 )
& ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4612_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L3: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4613_Icc__eq__Icc,axiom,
! [L: int,H2: int,L3: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H2 )
= ( set_or1266510415728281911st_int @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H2 )
& ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4614_Icc__eq__Icc,axiom,
! [L: real,H2: real,L3: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H2 )
= ( set_or1222579329274155063t_real @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H2 )
& ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4615_atLeastAtMost__iff,axiom,
! [I2: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4616_atLeastAtMost__iff,axiom,
! [I2: set_int,L: set_int,U: set_int] :
( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L @ U ) )
= ( ( ord_less_eq_set_int @ L @ I2 )
& ( ord_less_eq_set_int @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4617_atLeastAtMost__iff,axiom,
! [I2: rat,L: rat,U: rat] :
( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I2 )
& ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4618_atLeastAtMost__iff,axiom,
! [I2: num,L: num,U: num] :
( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I2 )
& ( ord_less_eq_num @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4619_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4620_atLeastAtMost__iff,axiom,
! [I2: int,L: int,U: int] :
( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I2 )
& ( ord_less_eq_int @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4621_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4622_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_4623_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_4624_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% take_bit_of_0
thf(fact_4625_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% take_bit_of_0
thf(fact_4626_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_4627_semiring__norm_I80_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(80)
thf(fact_4628_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_4629_neg__0__le__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_4630_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_4631_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4632_neg__le__0__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4633_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4634_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_4635_less__eq__neg__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_4636_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_4637_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4638_neg__less__eq__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4639_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4640_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_4641_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_4642_less__neg__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_4643_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_4644_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_4645_neg__less__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_pos
thf(fact_4646_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_4647_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_4648_neg__0__less__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_4649_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4650_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4651_neg__less__0__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4652_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_4653_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_4654_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_4655_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_4656_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_4657_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_4658_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_4659_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_4660_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4661_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4662_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4663_verit__minus__simplify_I3_J,axiom,
! [B: rat] :
( ( minus_minus_rat @ zero_zero_rat @ B )
= ( uminus_uminus_rat @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4664_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_4665_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_4666_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_4667_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_4668_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4669_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4670_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4671_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4672_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4673_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( minus_minus_rat @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4674_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4675_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4676_diff__minus__eq__add,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4677_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_4678_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_4679_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_4680_atLeastatMost__empty__iff2,axiom,
! [A: set_int,B: set_int] :
( ( bot_bot_set_set_int
= ( set_or370866239135849197et_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4681_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4682_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4683_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4684_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4685_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4686_atLeastatMost__empty__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( set_or370866239135849197et_int @ A @ B )
= bot_bot_set_set_int )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4687_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4688_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4689_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4690_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4691_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4692_atLeastatMost__subset__iff,axiom,
! [A: set_int,B: set_int,C2: set_int,D: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C2 @ A )
& ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4693_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C2 @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4694_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C2: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C2 @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4695_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4696_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C2 @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4697_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C2 @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4698_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_4699_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_4700_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_4701_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_4702_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_4703_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
= ( ord_less_rat @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_4704_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_4705_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_4706_take__bit__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% take_bit_0
thf(fact_4707_take__bit__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% take_bit_0
thf(fact_4708_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_4709_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_4710_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_4711_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_4712_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_4713_semiring__norm_I81_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(81)
thf(fact_4714_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_4715_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_4716_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_4717_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_4718_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_4719_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_4720_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_4721_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_4722_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_4723_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_4724_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_4725_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_4726_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_4727_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_4728_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4729_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4730_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4731_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4732_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4733_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4734_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4735_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4736_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4737_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_4738_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_4739_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_4740_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_4741_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4742_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4743_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4744_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4745_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4746_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4747_neg__numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_4748_neg__numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_4749_neg__numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_4750_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_4751_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_4752_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_4753_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4754_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4755_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4756_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4757_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4758_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4759_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4760_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4761_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4762_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4763_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4764_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4765_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4766_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_4767_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_4768_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_4769_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4770_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4771_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4772_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4773_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_4774_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_4775_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4776_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4777_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4778_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4779_even__take__bit__eq,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_4780_even__take__bit__eq,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_4781_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_4782_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_4783_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V: num] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_4784_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_4785_take__bit__Suc__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_4786_take__bit__Suc__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_4787_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_4788_take__bit__of__exp,axiom,
! [M2: nat,N: nat] :
( ( bit_se2925701944663578781it_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_4789_take__bit__of__exp,axiom,
! [M2: nat,N: nat] :
( ( bit_se2923211474154528505it_int @ M2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_4790_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_4791_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_4792_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_4793_compl__le__swap2,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_4794_compl__le__swap1,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
=> ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_4795_compl__mono,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% compl_mono
thf(fact_4796_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4797_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4798_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4799_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_4800_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_4801_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_4802_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_4803_take__bit__add,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
= ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% take_bit_add
thf(fact_4804_take__bit__add,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% take_bit_add
thf(fact_4805_take__bit__tightened,axiom,
! [N: nat,A: nat,B: nat,M2: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ A )
= ( bit_se2925701944663578781it_nat @ N @ B ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( bit_se2925701944663578781it_nat @ M2 @ A )
= ( bit_se2925701944663578781it_nat @ M2 @ B ) ) ) ) ).
% take_bit_tightened
thf(fact_4806_take__bit__tightened,axiom,
! [N: nat,A: int,B: int,M2: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ A )
= ( bit_se2923211474154528505it_int @ N @ B ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( bit_se2923211474154528505it_int @ M2 @ A )
= ( bit_se2923211474154528505it_int @ M2 @ B ) ) ) ) ).
% take_bit_tightened
thf(fact_4807_take__bit__nat__less__eq__self,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 ) ).
% take_bit_nat_less_eq_self
thf(fact_4808_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_4809_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_4810_le__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% le_minus_iff
thf(fact_4811_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_4812_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4813_minus__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4814_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4815_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4816_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4817_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4818_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_4819_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_4820_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_4821_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4822_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4823_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4824_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_4825_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_4826_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_4827_minus__mult__commute,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_mult_commute
thf(fact_4828_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_4829_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_4830_minus__mult__commute,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_mult_commute
thf(fact_4831_square__eq__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ A )
= ( times_times_complex @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% square_eq_iff
thf(fact_4832_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_4833_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_4834_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ A )
= ( times_times_rat @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% square_eq_iff
thf(fact_4835_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4836_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4837_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4838_group__cancel_Oneg1,axiom,
! [A4: int,K2: int,A: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4839_group__cancel_Oneg1,axiom,
! [A4: real,K2: real,A: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( uminus_uminus_real @ A4 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4840_group__cancel_Oneg1,axiom,
! [A4: rat,K2: rat,A: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( uminus_uminus_rat @ A4 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4841_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4842_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4843_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4844_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4845_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4846_minus__diff__commute,axiom,
! [B: rat,A: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4847_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_4848_minus__divide__left,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_4849_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_4850_minus__divide__divide,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ).
% minus_divide_divide
thf(fact_4851_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_4852_minus__divide__right,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_divide_right
thf(fact_4853_Diff__infinite__finite,axiom,
! [T3: set_int,S2: set_int] :
( ( finite_finite_int @ T3 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4854_Diff__infinite__finite,axiom,
! [T3: set_complex,S2: set_complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4855_Diff__infinite__finite,axiom,
! [T3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ T3 )
=> ( ~ ( finite6177210948735845034at_nat @ S2 )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4856_Diff__infinite__finite,axiom,
! [T3: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4857_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_4858_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_4859_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_4860_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_4861_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_4862_psubset__imp__ex__mem,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4863_psubset__imp__ex__mem,axiom,
! [A4: set_real,B5: set_real] :
( ( ord_less_set_real @ A4 @ B5 )
=> ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4864_psubset__imp__ex__mem,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B5 )
=> ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4865_psubset__imp__ex__mem,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4866_psubset__imp__ex__mem,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4867_zdvd__zdiffD,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M2 @ N ) )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_4868_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_4869_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_4870_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_4871_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_4872_take__bit__minus__small__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K2 ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_4873_infinite__Icc,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_4874_infinite__Icc,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_4875_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_4876_take__bit__tightened__less__eq__int,axiom,
! [M2: nat,N: nat,K2: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_4877_not__take__bit__negative,axiom,
! [N: nat,K2: int] :
~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ zero_zero_int ) ).
% not_take_bit_negative
thf(fact_4878_take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% take_bit_int_greater_self_iff
thf(fact_4879_signed__take__bit__take__bit,axiom,
! [M2: nat,N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( if_int_int @ ( ord_less_eq_nat @ N @ M2 ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M2 ) @ A ) ) ).
% signed_take_bit_take_bit
thf(fact_4880_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( P2 @ M3 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less
thf(fact_4881_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( P2 @ M3 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less
thf(fact_4882_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4883_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4884_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4885_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_4886_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_4887_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_4888_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_4889_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4890_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4891_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4892_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4893_not__numeral__less__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4894_not__numeral__less__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4895_not__numeral__less__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4896_neg__numeral__less__numeral,axiom,
! [M2: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_4897_neg__numeral__less__numeral,axiom,
! [M2: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_4898_neg__numeral__less__numeral,axiom,
! [M2: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_4899_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_4900_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_4901_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_4902_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_4903_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_4904_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_4905_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_4906_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_4907_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_4908_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_4909_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4910_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4911_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4912_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4913_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4914_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4915_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4916_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4917_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4918_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4919_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4920_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4921_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4922_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4923_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4924_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4925_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4926_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4927_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4928_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4929_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_4930_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_4931_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(4)
thf(fact_4932_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_4933_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_4934_less__minus__one__simps_I2_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% less_minus_one_simps(2)
thf(fact_4935_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_4936_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_4937_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_4938_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_4939_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_4940_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_4941_square__eq__1__iff,axiom,
! [X: complex] :
( ( ( times_times_complex @ X @ X )
= one_one_complex )
= ( ( X = one_one_complex )
| ( X
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_4942_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_4943_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_4944_square__eq__1__iff,axiom,
! [X: rat] :
( ( ( times_times_rat @ X @ X )
= one_one_rat )
= ( ( X = one_one_rat )
| ( X
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% square_eq_1_iff
thf(fact_4945_group__cancel_Osub2,axiom,
! [B5: int,K2: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4946_group__cancel_Osub2,axiom,
! [B5: real,K2: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K2 @ B ) )
=> ( ( minus_minus_real @ A @ B5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4947_group__cancel_Osub2,axiom,
! [B5: rat,K2: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K2 @ B ) )
=> ( ( minus_minus_rat @ A @ B5 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4948_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4949_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4950_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4952_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4953_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4954_take__bit__unset__bit__eq,axiom,
! [N: nat,M2: nat,A: nat] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
= ( bit_se2925701944663578781it_nat @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
= ( bit_se4205575877204974255it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_4955_take__bit__unset__bit__eq,axiom,
! [N: nat,M2: nat,A: int] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
= ( bit_se2923211474154528505it_int @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
= ( bit_se4203085406695923979it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_4956_take__bit__set__bit__eq,axiom,
! [N: nat,M2: nat,A: nat] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
= ( bit_se2925701944663578781it_nat @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
= ( bit_se7882103937844011126it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_4957_take__bit__set__bit__eq,axiom,
! [N: nat,M2: nat,A: int] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
= ( bit_se2923211474154528505it_int @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
= ( bit_se7879613467334960850it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_4958_take__bit__flip__bit__eq,axiom,
! [N: nat,M2: nat,A: nat] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
= ( bit_se2925701944663578781it_nat @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
= ( bit_se2161824704523386999it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_4959_take__bit__flip__bit__eq,axiom,
! [N: nat,M2: nat,A: int] :
( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
= ( bit_se2923211474154528505it_int @ N @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
= ( bit_se2159334234014336723it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_4960_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_4961_dvd__div__neg,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_4962_dvd__div__neg,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_4963_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_4964_dvd__neg__div,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_4965_dvd__neg__div,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_4966_subset__Compl__self__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% subset_Compl_self_eq
thf(fact_4967_subset__Compl__self__eq,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
= ( A4 = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_4968_subset__Compl__self__eq,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_4969_subset__Compl__self__eq,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_4970_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_4971_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_4972_int__le__induct,axiom,
! [I2: int,K2: int,P2: int > $o] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_4973_int__less__induct,axiom,
! [I2: int,K2: int,P2: int > $o] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P2 @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_4974_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_4975_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_4976_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_4977_take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_4978_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_4979_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_4980_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_4981_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_4982_atLeastatMost__psubset__iff,axiom,
! [A: set_int,B: set_int,C2: set_int,D: set_int] :
( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C2 @ A )
& ( ord_less_eq_set_int @ B @ D )
& ( ( ord_less_set_int @ C2 @ A )
| ( ord_less_set_int @ B @ D ) ) ) )
& ( ord_less_eq_set_int @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4983_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C2 @ A )
& ( ord_less_eq_rat @ B @ D )
& ( ( ord_less_rat @ C2 @ A )
| ( ord_less_rat @ B @ D ) ) ) )
& ( ord_less_eq_rat @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4984_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C2: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C2 @ A )
& ( ord_less_eq_num @ B @ D )
& ( ( ord_less_num @ C2 @ A )
| ( ord_less_num @ B @ D ) ) ) )
& ( ord_less_eq_num @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4985_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C2 @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4986_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C2 @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C2 @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4987_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C2 @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C2 @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4988_take__bit__signed__take__bit,axiom,
! [M2: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( bit_se2923211474154528505it_int @ M2 @ ( bit_ri631733984087533419it_int @ N @ A ) )
= ( bit_se2923211474154528505it_int @ M2 @ A ) ) ) ).
% take_bit_signed_take_bit
thf(fact_4989_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_4990_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_4991_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_4992_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4993_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4994_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4995_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_4996_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_4997_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_4998_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_4999_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5000_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5001_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_5002_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_5003_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_5004_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_5005_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_5006_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_5007_numeral__Bit1,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_5008_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_5009_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_5010_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_5011_numeral__Bit1,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_Bit1
thf(fact_5012_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_5013_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_5014_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_5015_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_5016_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_5017_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_5018_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_5019_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5020_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5021_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5022_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_5023_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_5024_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_5025_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_5026_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_5027_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_5028_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5029_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5030_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5031_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_5032_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_5033_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_5034_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5035_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5036_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5037_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5038_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5039_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5040_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_5041_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_5042_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_less_neg_one
thf(fact_5043_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_5044_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_5045_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_5046_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_5047_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_5048_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).
% neg_numeral_less_one
thf(fact_5049_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C2: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C2
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ( times_times_complex @ C2 @ B )
= ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5050_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C2: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C2
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C2 @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5051_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C2: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C2
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
= ( ( times_times_rat @ C2 @ B )
= ( uminus_uminus_rat @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5052_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C2: complex] :
( ( B != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= C2 )
= ( ( uminus1482373934393186551omplex @ A )
= ( times_times_complex @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5053_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C2: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C2 )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5054_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C2: rat] :
( ( B != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= C2 )
= ( ( uminus_uminus_rat @ A )
= ( times_times_rat @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5055_minus__divide__eq__eq,axiom,
! [B: complex,C2: complex,A: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B )
= ( times_times_complex @ A @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5056_minus__divide__eq__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5057_minus__divide__eq__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B )
= ( times_times_rat @ A @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5058_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C2: complex] :
( ( A
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A @ C2 )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5059_eq__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= ( uminus_uminus_real @ B ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5060_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A
= ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ A @ C2 )
= ( uminus_uminus_rat @ B ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5061_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B != zero_zero_complex )
& ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5062_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5063_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B != zero_zero_rat )
& ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5064_take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_5065_take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_5066_plusinfinity,axiom,
! [D: int,P5: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_5067_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_5068_int__induct,axiom,
! [P2: int > $o,K2: int,I2: int] :
( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_induct
thf(fact_5069_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_5070_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_5071_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_5072_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_5073_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).
% numeral_code(3)
thf(fact_5074_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_5075_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_5076_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_5077_take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_5078_less__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5079_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5080_minus__divide__less__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5081_minus__divide__less__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5082_neg__less__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5083_neg__less__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5084_neg__minus__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5085_neg__minus__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5086_pos__less__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5087_pos__less__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5088_pos__minus__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5089_pos__minus__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5090_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: complex,C2: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5091_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5092_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5093_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C2: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5094_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ( divide_divide_real @ B @ C2 )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5095_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C2 )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5096_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_5097_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_5098_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5099_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5100_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5101_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5102_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5103_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5104_minus__divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5105_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5106_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5107_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5108_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5109_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5110_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5111_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5112_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5113_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_5114_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_5115_signed__take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_5116_decr__mult__lemma,axiom,
! [D: int,P2: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_5117_mod__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K2 @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_5118_verit__less__mono__div__int2,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_5119_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_5120_take__bit__nat__eq__self,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2925701944663578781it_nat @ N @ M2 )
= M2 ) ) ).
% take_bit_nat_eq_self
thf(fact_5121_take__bit__nat__less__exp,axiom,
! [N: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_5122_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ M2 )
= M2 )
= ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_5123_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_5124_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_5125_take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).
% take_bit_int_less_self_iff
thf(fact_5126_take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_5127_le__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5128_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5129_minus__divide__le__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5130_minus__divide__le__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5131_neg__le__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5132_neg__le__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5133_neg__minus__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5134_neg__minus__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5135_pos__le__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5136_pos__le__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5137_pos__minus__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5138_pos__minus__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5139_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5140_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5141_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5142_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5143_cong__exp__iff__simps_I11_J,axiom,
! [M2: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5144_cong__exp__iff__simps_I11_J,axiom,
! [M2: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5145_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5146_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5147_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_5148_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5149_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5150_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5151_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5152_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_5153_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
=> ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_5154_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
& ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_5155_take__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ A )
= zero_zero_nat )
= ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_5156_take__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( ( bit_se2923211474154528505it_int @ N @ A )
= zero_zero_int )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_5157_take__bit__nat__less__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_5158_take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
= K2 )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_5159_take__bit__int__eq__self,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ K2 )
= K2 ) ) ) ).
% take_bit_int_eq_self
thf(fact_5160_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5161_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5162_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5163_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5164_square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_5165_square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% square_le_1
thf(fact_5166_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_5167_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_5168_div__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K2 @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_5169_take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).
% take_bit_int_greater_eq
thf(fact_5170_take__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5171_take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5172_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_5173_stable__imp__take__bit__eq,axiom,
! [A: nat,N: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5174_stable__imp__take__bit__eq,axiom,
! [A: int,N: nat] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5175_mod__exhaust__less__4,axiom,
! [M2: nat] :
( ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= zero_zero_nat )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= one_one_nat )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_5176_ln__2__less__1,axiom,
ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% ln_2_less_1
thf(fact_5177_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5178_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5179_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5180_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5181_Bolzano,axiom,
! [A: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B3: real,C: real] :
( ( P2 @ A3 @ B3 )
=> ( ( P2 @ B3 @ C )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( P2 @ A3 @ C ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_5182_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_5183_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_5184_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_5185_case__prod__conv,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5186_case__prod__conv,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5187_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5188_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5189_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5190_divmod__algorithm__code_I2_J,axiom,
! [M2: num] :
( ( unique5052692396658037445od_int @ M2 @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(2)
thf(fact_5191_divmod__algorithm__code_I2_J,axiom,
! [M2: num] :
( ( unique5055182867167087721od_nat @ M2 @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(2)
thf(fact_5192_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5193_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5194_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5195_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5196_divmod__algorithm__code_I5_J,axiom,
! [M2: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5197_divmod__algorithm__code_I5_J,axiom,
! [M2: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5198_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
@ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5199_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
@ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5200_prod_Ocase__distrib,axiom,
! [H2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
= ( produc6081775807080527818_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5201_prod_Ocase__distrib,axiom,
! [H2: $o > nat,F: nat > nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
= ( produc6842872674320459806at_nat
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5202_prod_Ocase__distrib,axiom,
! [H2: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
= ( produc6081775807080527818_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5203_prod_Ocase__distrib,axiom,
! [H2: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
= ( produc6842872674320459806at_nat
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5204_prod_Ocase__distrib,axiom,
! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5205_prod_Ocase__distrib,axiom,
! [H2: $o > product_prod_nat_nat > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5206_prod_Ocase__distrib,axiom,
! [H2: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5207_prod_Ocase__distrib,axiom,
! [H2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc6081775807080527818_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5208_prod_Ocase__distrib,axiom,
! [H2: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc6842872674320459806at_nat
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5209_prod_Ocase__distrib,axiom,
! [H2: $o > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
= ( produc27273713700761075at_nat
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5210_old_Oprod_Ocase,axiom,
! [F: nat > nat > $o,X1: nat,X2: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5211_old_Oprod_Ocase,axiom,
! [F: nat > nat > nat,X1: nat,X2: nat] :
( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5212_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5213_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5214_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X2: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5215_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q2 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc6081775807080527818_nat_o @ F @ P )
= ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_5216_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q2 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc6842872674320459806at_nat @ F @ P )
= ( produc6842872674320459806at_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_5217_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q2 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc27273713700761075at_nat @ F @ P )
= ( produc27273713700761075at_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_5218_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q2 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc8739625826339149834_nat_o @ F @ P )
= ( produc8739625826339149834_nat_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_5219_split__cong,axiom,
! [Q2: product_prod_int_int,F: int > int > $o,G: int > int > $o,P: product_prod_int_int] :
( ! [X4: int,Y3: int] :
( ( ( product_Pair_int_int @ X4 @ Y3 )
= Q2 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc4947309494688390418_int_o @ F @ P )
= ( produc4947309494688390418_int_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_5220_case__prodE2,axiom,
! [Q: $o > $o,P2: nat > nat > $o,Z: product_prod_nat_nat] :
( ( Q @ ( produc6081775807080527818_nat_o @ P2 @ Z ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5221_case__prodE2,axiom,
! [Q: nat > $o,P2: nat > nat > nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc6842872674320459806at_nat @ P2 @ Z ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5222_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc27273713700761075at_nat @ P2 @ Z ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5223_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
( ( Q @ ( produc8739625826339149834_nat_o @ P2 @ Z ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5224_case__prodE2,axiom,
! [Q: $o > $o,P2: int > int > $o,Z: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P2 @ Z ) )
=> ~ ! [X4: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5225_case__prod__eta,axiom,
! [F: product_prod_nat_nat > $o] :
( ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5226_case__prod__eta,axiom,
! [F: product_prod_nat_nat > nat] :
( ( produc6842872674320459806at_nat
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5227_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5228_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5229_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X5: int,Y5: int] : ( F @ ( product_Pair_int_int @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5230_cond__case__prod__eta,axiom,
! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc6081775807080527818_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5231_cond__case__prod__eta,axiom,
! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc6842872674320459806at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5232_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5233_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5234_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X4: int,Y3: int] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5235_periodic__finite__ex,axiom,
! [D: int,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P2 @ X4 )
= ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P2 @ X5 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_5236_aset_I7_J,axiom,
! [D6: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X3 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_5237_aset_I5_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A4 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X3 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X3 @ D6 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_5238_aset_I4_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A4 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 != T )
=> ( ( plus_plus_int @ X3 @ D6 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_5239_aset_I3_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 = T )
=> ( ( plus_plus_int @ X3 @ D6 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_5240_bset_I7_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X3 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_5241_bset_I5_J,axiom,
! [D6: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X3 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X3 @ D6 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_5242_bset_I4_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 != T )
=> ( ( minus_minus_int @ X3 @ D6 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_5243_bset_I3_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 = T )
=> ( ( minus_minus_int @ X3 @ D6 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_5244_bset_I6_J,axiom,
! [D6: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X3 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D6 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_5245_bset_I8_J,axiom,
! [D6: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X3 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_5246_aset_I6_J,axiom,
! [D6: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X3 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D6 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_5247_aset_I8_J,axiom,
! [D6: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A4 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X3 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_5248_cpmi,axiom,
! [D6: int,P2: int > $o,P5: int > $o,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D6 ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( P2 @ ( plus_plus_int @ Y5 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_5249_cppi,axiom,
! [D6: int,P2: int > $o,P5: int > $o,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D6 ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ A4 )
& ( P2 @ ( minus_minus_int @ Y5 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_5250_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_5251_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_5252_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5253_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5254_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5255_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5256_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( tanh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% tanh_ln_real
thf(fact_5257_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M3: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M3 @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
@ ( produc2626176000494625587at_nat
@ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5258_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] :
( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_5259_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] :
( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_5260_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] :
( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_5261_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] :
( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_5262_take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_numeral_bit1
thf(fact_5263_take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_bit1
thf(fact_5264_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_5265_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5266_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5267_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5268_of__int__eq__iff,axiom,
! [W2: int,Z: int] :
( ( ( ring_1_of_int_real @ W2 )
= ( ring_1_of_int_real @ Z ) )
= ( W2 = Z ) ) ).
% of_int_eq_iff
thf(fact_5269_of__int__eq__iff,axiom,
! [W2: int,Z: int] :
( ( ( ring_1_of_int_rat @ W2 )
= ( ring_1_of_int_rat @ Z ) )
= ( W2 = Z ) ) ).
% of_int_eq_iff
thf(fact_5270_case__prodI2,axiom,
! [P: produc859450856879609959at_nat,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( P
= ( produc6161850002892822231at_nat @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc6590410687421337004_nat_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5271_case__prodI2,axiom,
! [P: produc1319942482725812455at_nat,C2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
( ( P
= ( produc9060074326276436823at_nat @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc9020218426428501292_nat_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5272_case__prodI2,axiom,
! [P: produc3843707927480180839at_nat,C2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc2922128104949294807at_nat @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc410239310623530412_nat_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5273_case__prodI2,axiom,
! [P: product_prod_nat_nat,C2: nat > nat > $o] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc6081775807080527818_nat_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5274_case__prodI2,axiom,
! [P: product_prod_int_int,C2: int > int > $o] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc4947309494688390418_int_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5275_case__prodI,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( F @ A @ B )
=> ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5276_case__prodI,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( F @ A @ B )
=> ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5277_case__prodI,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( F @ A @ B )
=> ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5278_case__prodI,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( F @ A @ B )
=> ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5279_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5280_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: real,C2: nat > nat > set_real] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_real @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_real @ Z @ ( produc3668448655016342576t_real @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5281_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: nat,C2: nat > nat > set_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5282_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: int,C2: nat > nat > set_int] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_int @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_int @ Z @ ( produc2011625207790711856et_int @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5283_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: real,C2: int > int > set_real] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_real @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_real @ Z @ ( produc6452406959799940328t_real @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5284_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: nat,C2: int > int > set_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5285_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: int,C2: int > int > set_int] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_int @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_int @ Z @ ( produc73460835934605544et_int @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5286_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: set_nat,C2: nat > nat > set_set_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_set_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5287_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: set_nat,C2: int > int > set_set_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_set_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5288_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5289_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: product_prod_nat_nat,C2: int > int > set_Pr1261947904930325089at_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5290_mem__case__prodI,axiom,
! [Z: real,C2: nat > nat > set_real,A: nat,B: nat] :
( ( member_real @ Z @ ( C2 @ A @ B ) )
=> ( member_real @ Z @ ( produc3668448655016342576t_real @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5291_mem__case__prodI,axiom,
! [Z: nat,C2: nat > nat > set_nat,A: nat,B: nat] :
( ( member_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5292_mem__case__prodI,axiom,
! [Z: int,C2: nat > nat > set_int,A: nat,B: nat] :
( ( member_int @ Z @ ( C2 @ A @ B ) )
=> ( member_int @ Z @ ( produc2011625207790711856et_int @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5293_mem__case__prodI,axiom,
! [Z: real,C2: int > int > set_real,A: int,B: int] :
( ( member_real @ Z @ ( C2 @ A @ B ) )
=> ( member_real @ Z @ ( produc6452406959799940328t_real @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5294_mem__case__prodI,axiom,
! [Z: nat,C2: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5295_mem__case__prodI,axiom,
! [Z: int,C2: int > int > set_int,A: int,B: int] :
( ( member_int @ Z @ ( C2 @ A @ B ) )
=> ( member_int @ Z @ ( produc73460835934605544et_int @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5296_mem__case__prodI,axiom,
! [Z: set_nat,C2: nat > nat > set_set_nat,A: nat,B: nat] :
( ( member_set_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5297_mem__case__prodI,axiom,
! [Z: set_nat,C2: int > int > set_set_nat,A: int,B: int] :
( ( member_set_nat @ Z @ ( C2 @ A @ B ) )
=> ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5298_mem__case__prodI,axiom,
! [Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5299_mem__case__prodI,axiom,
! [Z: product_prod_nat_nat,C2: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5300_case__prodI2_H,axiom,
! [P: product_prod_nat_nat,C2: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B3 )
= P )
=> ( C2 @ A3 @ B3 @ X ) )
=> ( produc8739625826339149834_nat_o @ C2 @ P @ X ) ) ).
% case_prodI2'
thf(fact_5301_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_5302_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n3304061248610475627l_real @ P2 ) ) ).
% of_int_of_bool
thf(fact_5303_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2052037380579107095ol_rat @ P2 ) ) ).
% of_int_of_bool
thf(fact_5304_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% of_int_of_bool
thf(fact_5305_tanh__0,axiom,
( ( tanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% tanh_0
thf(fact_5306_tanh__0,axiom,
( ( tanh_real @ zero_zero_real )
= zero_zero_real ) ).
% tanh_0
thf(fact_5307_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_5308_of__int__0,axiom,
( ( ring_17405671764205052669omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_int_0
thf(fact_5309_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_5310_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_5311_of__int__0,axiom,
( ( ring_1_of_int_rat @ zero_zero_int )
= zero_zero_rat ) ).
% of_int_0
thf(fact_5312_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_complex
= ( ring_17405671764205052669omplex @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_5313_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_5314_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_5315_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_rat
= ( ring_1_of_int_rat @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_5316_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_17405671764205052669omplex @ Z )
= zero_zero_complex )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_5317_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_5318_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_5319_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= zero_zero_rat )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_5320_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_5321_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_5322_of__int__le__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% of_int_le_iff
thf(fact_5323_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_rat @ Z )
= ( numeral_numeral_rat @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5324_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_int @ Z )
= ( numeral_numeral_int @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5325_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ( ring_1_of_int_real @ Z )
= ( numeral_numeral_real @ N ) )
= ( Z
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5326_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_rat @ K2 ) ) ).
% of_int_numeral
thf(fact_5327_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ K2 ) ) ).
% of_int_numeral
thf(fact_5328_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_real @ K2 ) ) ).
% of_int_numeral
thf(fact_5329_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_5330_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_5331_of__int__less__iff,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% of_int_less_iff
thf(fact_5332_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_5333_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_17405671764205052669omplex @ Z )
= one_one_complex )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_5334_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_5335_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= one_one_rat )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_5336_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_5337_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_5338_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_5339_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_5340_of__int__mult,axiom,
! [W2: int,Z: int] :
( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z ) )
= ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% of_int_mult
thf(fact_5341_of__int__mult,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_5342_of__int__mult,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_mult
thf(fact_5343_of__int__mult,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_5344_of__int__add,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_5345_of__int__add,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_5346_of__int__add,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_add
thf(fact_5347_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_5348_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_minus
thf(fact_5349_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_minus
thf(fact_5350_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_5351_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% real_sqrt_lt_0_iff
thf(fact_5352_of__int__diff,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_5353_of__int__diff,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_5354_of__int__diff,axiom,
! [W2: int,Z: int] :
( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z ) )
= ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_diff
thf(fact_5355_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ).
% real_sqrt_lt_1_iff
thf(fact_5356_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_5357_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_5358_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_rat @ X )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_5359_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_5360_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_17405671764205052669omplex @ X )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_5361_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_5362_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
= ( ring_1_of_int_rat @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_5363_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_5364_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
= ( ring_17405671764205052669omplex @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_5365_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_5366_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% of_int_power
thf(fact_5367_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% of_int_power
thf(fact_5368_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% of_int_power
thf(fact_5369_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% of_int_power
thf(fact_5370_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% tanh_real_neg_iff
thf(fact_5371_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% tanh_real_pos_iff
thf(fact_5372_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_5373_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_5374_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_5375_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_5376_max__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% max_Suc_numeral
thf(fact_5377_max__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_5378_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_5379_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_5380_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_5381_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_5382_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_5383_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_5384_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_5385_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_5386_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_5387_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_5388_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_5389_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_5390_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_5391_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_5392_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_5393_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5394_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5395_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5396_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5397_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5398_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5399_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_5400_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_5401_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_5402_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_5403_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_5404_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_5405_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_5406_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_5407_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_5408_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_5409_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_5410_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_5411_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_5412_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_5413_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_5414_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_5415_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_5416_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_5417_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_5418_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_5419_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_5420_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5421_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5422_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5423_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5424_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5425_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5426_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5427_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5428_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_5429_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_5430_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_5431_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_5432_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_5433_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_5434_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5435_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5436_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5437_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5438_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5439_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5440_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5441_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5442_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5443_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5444_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5445_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5446_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5447_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5448_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5449_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5450_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5451_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5452_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5453_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5454_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5455_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5456_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5457_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5458_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5459_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5460_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5461_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5462_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5463_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_5464_ex__le__of__int,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% ex_le_of_int
thf(fact_5465_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_5466_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_less_of_int
thf(fact_5467_ex__less__of__int,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).
% ex_less_of_int
thf(fact_5468_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_5469_ex__of__int__less,axiom,
! [X: rat] :
? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_5470_mult__of__int__commute,axiom,
! [X: int,Y: complex] :
( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% mult_of_int_commute
thf(fact_5471_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_5472_mult__of__int__commute,axiom,
! [X: int,Y: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% mult_of_int_commute
thf(fact_5473_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_5474_mem__case__prodE,axiom,
! [Z: real,C2: nat > nat > set_real,P: product_prod_nat_nat] :
( ( member_real @ Z @ ( produc3668448655016342576t_real @ C2 @ P ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( member_real @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5475_mem__case__prodE,axiom,
! [Z: nat,C2: nat > nat > set_nat,P: product_prod_nat_nat] :
( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C2 @ P ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5476_mem__case__prodE,axiom,
! [Z: int,C2: nat > nat > set_int,P: product_prod_nat_nat] :
( ( member_int @ Z @ ( produc2011625207790711856et_int @ C2 @ P ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5477_mem__case__prodE,axiom,
! [Z: real,C2: int > int > set_real,P: product_prod_int_int] :
( ( member_real @ Z @ ( produc6452406959799940328t_real @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_real @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5478_mem__case__prodE,axiom,
! [Z: nat,C2: int > int > set_nat,P: product_prod_int_int] :
( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5479_mem__case__prodE,axiom,
! [Z: int,C2: int > int > set_int,P: product_prod_int_int] :
( ( member_int @ Z @ ( produc73460835934605544et_int @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5480_mem__case__prodE,axiom,
! [Z: set_nat,C2: nat > nat > set_set_nat,P: product_prod_nat_nat] :
( ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C2 @ P ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( member_set_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5481_mem__case__prodE,axiom,
! [Z: set_nat,C2: int > int > set_set_nat,P: product_prod_int_int] :
( ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_set_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5482_mem__case__prodE,axiom,
! [Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ P ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5483_mem__case__prodE,axiom,
! [Z: product_prod_nat_nat,C2: int > int > set_Pr1261947904930325089at_nat,P: product_prod_int_int] :
( ( member8440522571783428010at_nat @ Z @ ( produc1656060378719767003at_nat @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5484_of__int__max,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
= ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_max
thf(fact_5485_of__int__max,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_rat @ ( ord_max_int @ X @ Y ) )
= ( ord_max_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y ) ) ) ).
% of_int_max
thf(fact_5486_of__int__max,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
= ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_max
thf(fact_5487_case__prodE,axiom,
! [C2: product_prod_nat_nat > product_prod_nat_nat > $o,P: produc859450856879609959at_nat] :
( ( produc6590410687421337004_nat_o @ C2 @ P )
=> ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( P
= ( produc6161850002892822231at_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5488_case__prodE,axiom,
! [C2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,P: produc1319942482725812455at_nat] :
( ( produc9020218426428501292_nat_o @ C2 @ P )
=> ~ ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( P
= ( produc9060074326276436823at_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5489_case__prodE,axiom,
! [C2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P: produc3843707927480180839at_nat] :
( ( produc410239310623530412_nat_o @ C2 @ P )
=> ~ ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc2922128104949294807at_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5490_case__prodE,axiom,
! [C2: nat > nat > $o,P: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C2 @ P )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5491_case__prodE,axiom,
! [C2: int > int > $o,P: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C2 @ P )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5492_case__prodD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5493_case__prodD,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5494_case__prodD,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5495_case__prodD,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5496_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5497_case__prodE_H,axiom,
! [C2: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C2 @ P @ Z )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_5498_case__prodD_H,axiom,
! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C2 )
=> ( R @ A @ B @ C2 ) ) ).
% case_prodD'
thf(fact_5499_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_5500_Collect__case__prod__mono,axiom,
! [A4: nat > nat > $o,B5: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A4 @ B5 )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5501_Collect__case__prod__mono,axiom,
! [A4: int > int > $o,B5: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5502_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% tanh_real_lt_1
thf(fact_5503_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_5504_sqrt2__less__2,axiom,
ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% sqrt2_less_2
thf(fact_5505_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_5506_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_nonneg
thf(fact_5507_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_5508_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_5509_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_pos
thf(fact_5510_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_5511_floor__exists1,axiom,
! [X: real] :
? [X4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X4 ) ) ) ).
% floor_exists1
thf(fact_5512_floor__exists1,axiom,
! [X: rat] :
? [X4: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X4 ) ) ) ).
% floor_exists1
thf(fact_5513_floor__exists,axiom,
! [X: real] :
? [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_5514_floor__exists,axiom,
! [X: rat] :
? [Z3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_5515_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_5516_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_5517_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_5518_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_5519_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_5520_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_5521_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_5522_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_5523_ln__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( sqrt @ X ) )
= ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% ln_sqrt
thf(fact_5524_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_5525_arsinh__real__aux,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% arsinh_real_aux
thf(fact_5526_round__unique,axiom,
! [X: real,Y: int] :
( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= Y ) ) ) ).
% round_unique
thf(fact_5527_round__unique,axiom,
! [X: rat,Y: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y ) ) ) ).
% round_unique
thf(fact_5528_of__int__round__gt,axiom,
! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_gt
thf(fact_5529_of__int__round__gt,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_gt
thf(fact_5530_of__int__round__ge,axiom,
! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_ge
thf(fact_5531_of__int__round__ge,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_ge
thf(fact_5532_of__int__round__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_5533_of__int__round__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_5534_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z5: int] :
( ( ord_less_eq_int @ D5 @ Z6 )
& ( ord_less_int @ Z6 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5535_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z5: int] :
( ( ord_less_eq_int @ D5 @ Z5 )
& ( ord_less_int @ Z6 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5536_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ M2 @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ M2 @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_5537_split__part,axiom,
! [P2: $o,Q: nat > nat > $o] :
( ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] :
( P2
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod_nat_nat] :
( P2
& ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_5538_split__part,axiom,
! [P2: $o,Q: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( P2
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P2
& ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_5539_sum_Oneutral__const,axiom,
! [A4: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu3: int] : zero_zero_int
@ A4 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_5540_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu3: nat] : zero_zero_nat
@ A4 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_5541_sum_Oneutral__const,axiom,
! [A4: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A4 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_5542_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu3: nat] : zero_zero_real
@ A4 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_5543_sum_Oempty,axiom,
! [G: real > complex] :
( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
= zero_zero_complex ) ).
% sum.empty
thf(fact_5544_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_5545_sum_Oempty,axiom,
! [G: real > rat] :
( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
= zero_zero_rat ) ).
% sum.empty
thf(fact_5546_sum_Oempty,axiom,
! [G: real > nat] :
( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_5547_sum_Oempty,axiom,
! [G: real > int] :
( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_5548_sum_Oempty,axiom,
! [G: nat > complex] :
( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
= zero_zero_complex ) ).
% sum.empty
thf(fact_5549_sum_Oempty,axiom,
! [G: nat > rat] :
( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_5550_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_5551_sum_Oempty,axiom,
! [G: int > complex] :
( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
= zero_zero_complex ) ).
% sum.empty
thf(fact_5552_sum_Oempty,axiom,
! [G: int > real] :
( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
= zero_zero_real ) ).
% sum.empty
thf(fact_5553_sum__eq__0__iff,axiom,
! [F3: set_int,F: int > nat] :
( ( finite_finite_int @ F3 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ F3 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_5554_sum__eq__0__iff,axiom,
! [F3: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ F3 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_5555_sum__eq__0__iff,axiom,
! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( ( groups977919841031483927at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ F3 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_5556_sum__eq__0__iff,axiom,
! [F3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ F3 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_5557_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups2073611262835488442omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_5558_sum_Oinfinite,axiom,
! [A4: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_5559_sum_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_5560_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_5561_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_5562_sum_Oinfinite,axiom,
! [A4: set_int,G: int > rat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_5563_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_5564_sum_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_5565_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_5566_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_5567_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_5568_round__0,axiom,
( ( archim7778729529865785530nd_rat @ zero_zero_rat )
= zero_zero_int ) ).
% round_0
thf(fact_5569_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_5570_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_5571_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_5572_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_5573_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_5574_sum_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_5575_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_5576_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_5577_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_5578_sum_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_5579_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_5580_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_5581_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_5582_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_5583_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_5584_sum_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_5585_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_5586_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_5587_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_5588_sum_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_5589_of__int__sum,axiom,
! [F: complex > int,A4: set_complex] :
( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A4 ) )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( ring_17405671764205052669omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_5590_of__int__sum,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A4 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_5591_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_5592_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X5: int] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_5593_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_5594_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_nat_nat] :
( produc6081775807080527818_nat_o
@ ^ [Uu3: nat,Uv3: nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5595_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] :
( produc4947309494688390418_int_o
@ ^ [Uu3: int,Uv3: int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5596_sum_Oneutral,axiom,
! [A4: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_5597_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_5598_sum_Oneutral,axiom,
! [A4: set_complex,G: complex > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_5599_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_5600_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A4: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5601_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A4: set_nat] :
( ( ( groups2073611262835488442omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5602_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A4: set_int] :
( ( ( groups3049146728041665814omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5603_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A4: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5604_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A4: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5605_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A4: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5606_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A4: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5607_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A4: set_int] :
( ( ( groups3906332499630173760nt_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5608_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A4: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5609_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A4: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_5610_sum__mono,axiom,
! [K5: set_real,F: real > rat,G: real > rat] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5611_sum__mono,axiom,
! [K5: set_nat,F: nat > rat,G: nat > rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5612_sum__mono,axiom,
! [K5: set_int,F: int > rat,G: int > rat] :
( ! [I3: int] :
( ( member_int @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5613_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5614_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5615_sum__mono,axiom,
! [K5: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5616_sum__mono,axiom,
! [K5: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5617_sum__mono,axiom,
! [K5: set_int,F: int > int,G: int > int] :
( ! [I3: int] :
( ( member_int @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5618_sum__mono,axiom,
! [K5: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5619_sum__mono,axiom,
! [K5: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K5 ) @ ( groups6591440286371151544t_real @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5620_sum_Odistrib,axiom,
! [G: int > int,H2: int > int,A4: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( plus_plus_int @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5621_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A4: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( plus_plus_nat @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5622_sum_Odistrib,axiom,
! [G: complex > complex,H2: complex > complex,A4: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( plus_plus_complex @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5623_sum_Odistrib,axiom,
! [G: nat > real,H2: nat > real,A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( plus_plus_real @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5624_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups1932886352136224148al_int
@ ^ [X5: real] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups1932886352136224148al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5625_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_int,G: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups3539618377306564664at_int
@ ^ [X5: nat] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups3539618377306564664at_int
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5626_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups5690904116761175830ex_int
@ ^ [X5: complex] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups5690904116761175830ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5627_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1935376822645274424al_nat
@ ^ [X5: real] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups1935376822645274424al_nat
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5628_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X5: int] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5629_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X5: complex] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups5693394587270226106ex_nat
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5630_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_complex,G: real > complex > complex,R: real > complex > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups5754745047067104278omplex
@ ^ [X5: real] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups5754745047067104278omplex
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5631_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups2073611262835488442omplex
@ ^ [X5: nat] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups2073611262835488442omplex
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5632_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_complex,G: int > complex > complex,R: int > complex > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups3049146728041665814omplex
@ ^ [X5: int] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups3049146728041665814omplex
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5633_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > real,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups8097168146408367636l_real
@ ^ [X5: real] :
( groups6591440286371151544t_real @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups6591440286371151544t_real
@ ^ [Y5: nat] :
( groups8097168146408367636l_real
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5634_sum__nonpos,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5635_sum__nonpos,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5636_sum__nonpos,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5637_sum__nonpos,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5638_sum__nonpos,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5639_sum__nonpos,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5640_sum__nonpos,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5641_sum__nonpos,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_5642_sum__nonpos,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_5643_sum__nonpos,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_5644_sum__nonneg,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5645_sum__nonneg,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5646_sum__nonneg,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5647_sum__nonneg,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5648_sum__nonneg,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5649_sum__nonneg,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5650_sum__nonneg,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5651_sum__nonneg,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5652_sum__nonneg,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5653_sum__nonneg,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5654_sum__mono__inv,axiom,
! [F: real > rat,I6: set_real,G: real > rat,I2: real] :
( ( ( groups1300246762558778688al_rat @ F @ I6 )
= ( groups1300246762558778688al_rat @ G @ I6 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5655_sum__mono__inv,axiom,
! [F: nat > rat,I6: set_nat,G: nat > rat,I2: nat] :
( ( ( groups2906978787729119204at_rat @ F @ I6 )
= ( groups2906978787729119204at_rat @ G @ I6 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I6 )
=> ( ( finite_finite_nat @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5656_sum__mono__inv,axiom,
! [F: int > rat,I6: set_int,G: int > rat,I2: int] :
( ( ( groups3906332499630173760nt_rat @ F @ I6 )
= ( groups3906332499630173760nt_rat @ G @ I6 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I2 @ I6 )
=> ( ( finite_finite_int @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5657_sum__mono__inv,axiom,
! [F: complex > rat,I6: set_complex,G: complex > rat,I2: complex] :
( ( ( groups5058264527183730370ex_rat @ F @ I6 )
= ( groups5058264527183730370ex_rat @ G @ I6 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5658_sum__mono__inv,axiom,
! [F: real > nat,I6: set_real,G: real > nat,I2: real] :
( ( ( groups1935376822645274424al_nat @ F @ I6 )
= ( groups1935376822645274424al_nat @ G @ I6 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5659_sum__mono__inv,axiom,
! [F: int > nat,I6: set_int,G: int > nat,I2: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I6 )
= ( groups4541462559716669496nt_nat @ G @ I6 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I2 @ I6 )
=> ( ( finite_finite_int @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5660_sum__mono__inv,axiom,
! [F: complex > nat,I6: set_complex,G: complex > nat,I2: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I6 )
= ( groups5693394587270226106ex_nat @ G @ I6 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5661_sum__mono__inv,axiom,
! [F: real > int,I6: set_real,G: real > int,I2: real] :
( ( ( groups1932886352136224148al_int @ F @ I6 )
= ( groups1932886352136224148al_int @ G @ I6 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5662_sum__mono__inv,axiom,
! [F: nat > int,I6: set_nat,G: nat > int,I2: nat] :
( ( ( groups3539618377306564664at_int @ F @ I6 )
= ( groups3539618377306564664at_int @ G @ I6 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I6 )
=> ( ( finite_finite_nat @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5663_sum__mono__inv,axiom,
! [F: complex > int,I6: set_complex,G: complex > int,I2: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I6 )
= ( groups5690904116761175830ex_int @ G @ I6 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5664_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > complex,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups5754745047067104278omplex @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5754745047067104278omplex
@ ^ [X5: real] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5665_sum_Ointer__filter,axiom,
! [A4: set_nat,G: nat > complex,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups2073611262835488442omplex @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2073611262835488442omplex
@ ^ [X5: nat] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5666_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > complex,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups3049146728041665814omplex
@ ^ [X5: int] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5667_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X5: real] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5668_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5669_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X5: complex] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5670_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > rat,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups1300246762558778688al_rat
@ ^ [X5: real] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5671_sum_Ointer__filter,axiom,
! [A4: set_nat,G: nat > rat,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [X5: nat] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5672_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > rat,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups3906332499630173760nt_rat
@ ^ [X5: int] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5673_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > rat,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5058264527183730370ex_rat
@ ^ [X5: complex] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5674_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5675_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5676_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5677_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5678_sum__nonneg__eq__0__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5679_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5680_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5681_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5682_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5683_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5684_sum__le__included,axiom,
! [S: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5685_sum__le__included,axiom,
! [S: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5686_sum__le__included,axiom,
! [S: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5687_sum__le__included,axiom,
! [S: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5688_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5689_sum__le__included,axiom,
! [S: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5690_sum__le__included,axiom,
! [S: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5691_sum__le__included,axiom,
! [S: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5692_sum__le__included,axiom,
! [S: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5693_sum__le__included,axiom,
! [S: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5694_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5695_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5696_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5697_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5698_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5699_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5700_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5701_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5702_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5703_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5704_sum_Orelated,axiom,
! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5705_sum_Orelated,axiom,
! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5706_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5707_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5708_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5709_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5710_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5711_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5712_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5713_sum_Orelated,axiom,
! [R: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X16: int,Y15: int,X22: int,Y23: int] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5714_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5715_sum__strict__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5716_sum__strict__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5717_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5718_sum__strict__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5719_sum__strict__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5720_sum__strict__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5721_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5722_sum__strict__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5723_sum__strict__mono,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5724_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5725_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5726_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5727_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5728_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5729_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I2: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5730_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_complex,S2: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5731_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I2: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5732_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I2: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5733_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_complex,S2: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5734_round__mono,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% round_mono
thf(fact_5735_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > real,B5: real,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= B5 )
=> ( ( member_real @ I2 @ S )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5736_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > real,B5: real,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S )
= B5 )
=> ( ( member_int @ I2 @ S )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5737_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > real,B5: real,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S )
= B5 )
=> ( ( member_complex @ I2 @ S )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5738_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > rat,B5: rat,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S )
= B5 )
=> ( ( member_real @ I2 @ S )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5739_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > rat,B5: rat,I2: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S )
= B5 )
=> ( ( member_nat @ I2 @ S )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5740_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > rat,B5: rat,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S )
= B5 )
=> ( ( member_int @ I2 @ S )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5741_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > rat,B5: rat,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S )
= B5 )
=> ( ( member_complex @ I2 @ S )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5742_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > nat,B5: nat,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= B5 )
=> ( ( member_real @ I2 @ S )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5743_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: int > nat,B5: nat,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S )
= B5 )
=> ( ( member_int @ I2 @ S )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5744_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: complex > nat,B5: nat,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S )
= B5 )
=> ( ( member_complex @ I2 @ S )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5745_sum__nonneg__0,axiom,
! [S: set_real,F: real > real,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= zero_zero_real )
=> ( ( member_real @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5746_sum__nonneg__0,axiom,
! [S: set_int,F: int > real,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S )
= zero_zero_real )
=> ( ( member_int @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5747_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > real,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S )
= zero_zero_real )
=> ( ( member_complex @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5748_sum__nonneg__0,axiom,
! [S: set_real,F: real > rat,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S )
= zero_zero_rat )
=> ( ( member_real @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5749_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > rat,I2: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S )
= zero_zero_rat )
=> ( ( member_nat @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5750_sum__nonneg__0,axiom,
! [S: set_int,F: int > rat,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S )
= zero_zero_rat )
=> ( ( member_int @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5751_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > rat,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S )
= zero_zero_rat )
=> ( ( member_complex @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5752_sum__nonneg__0,axiom,
! [S: set_real,F: real > nat,I2: real] :
( ( finite_finite_real @ S )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_real @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5753_sum__nonneg__0,axiom,
! [S: set_int,F: int > nat,I2: int] :
( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_int @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5754_sum__nonneg__0,axiom,
! [S: set_complex,F: complex > nat,I2: complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_complex @ I2 @ S )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5755_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( groups5754745047067104278omplex @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_complex ) ) ) )
= ( groups5754745047067104278omplex @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5756_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_complex ) ) ) )
= ( groups3049146728041665814omplex @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5757_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups8097168146408367636l_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5758_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups8778361861064173332t_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5759_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5760_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5761_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5762_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5763_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1935376822645274424al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_nat ) ) ) )
= ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5764_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5765_sum__pos2,axiom,
! [I6: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5766_sum__pos2,axiom,
! [I6: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I2 @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5767_sum__pos2,axiom,
! [I6: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5768_sum__pos2,axiom,
! [I6: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5769_sum__pos2,axiom,
! [I6: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I2 @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5770_sum__pos2,axiom,
! [I6: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I2 @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5771_sum__pos2,axiom,
! [I6: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5772_sum__pos2,axiom,
! [I6: set_real,I2: real,F: real > nat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5773_sum__pos2,axiom,
! [I6: set_int,I2: int,F: int > nat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I2 @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5774_sum__pos2,axiom,
! [I6: set_complex,I2: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5775_sum__pos,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5776_sum__pos,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5777_sum__pos,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5778_sum__pos,axiom,
! [I6: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5779_sum__pos,axiom,
! [I6: set_real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5780_sum__pos,axiom,
! [I6: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5781_sum__pos,axiom,
! [I6: set_int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5782_sum__pos,axiom,
! [I6: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5783_sum__pos,axiom,
! [I6: set_real,F: real > nat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5784_sum__pos,axiom,
! [I6: set_int,F: int > nat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5785_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ T3 )
= ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5786_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T3 )
= ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5787_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ T3 )
= ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5788_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ T3 )
= ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5789_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T3 )
= ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5790_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T3 )
= ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5791_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T3 )
= ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5792_sum_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ T3 )
= ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5793_sum_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ T3 )
= ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5794_sum_Omono__neutral__cong__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ T3 )
= ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5795_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5796_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5797_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5798_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ S2 )
= ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5799_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5800_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5801_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5802_sum_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ S2 )
= ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5803_sum_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5804_sum_Omono__neutral__cong__left,axiom,
! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5805_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ T3 )
= ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5806_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T3 )
= ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5807_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T3 )
= ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5808_sum_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ T3 )
= ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5809_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ T3 )
= ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5810_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ T3 )
= ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5811_sum_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ T3 )
= ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5812_sum_Omono__neutral__right,axiom,
! [T3: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ T3 )
= ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5813_sum_Omono__neutral__right,axiom,
! [T3: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ T3 )
= ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5814_sum_Omono__neutral__right,axiom,
! [T3: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ T3 )
= ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5815_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5816_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5817_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5818_sum_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5819_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5820_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ S2 )
= ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5821_sum_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5822_sum_Omono__neutral__left,axiom,
! [T3: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5823_sum_Omono__neutral__left,axiom,
! [T3: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5824_sum_Omono__neutral__left,axiom,
! [T3: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ S2 )
= ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5825_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) )
=> ( ( groups5754745047067104278omplex @ G @ A4 )
= ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5826_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) )
=> ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5827_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5828_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) )
=> ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5829_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5830_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5831_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5832_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) )
=> ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5833_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5834_sum_Osame__carrierI,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) )
=> ( ( groups2073611262835488442omplex @ G @ A4 )
= ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5835_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ A4 )
= ( groups5754745047067104278omplex @ H2 @ B5 ) )
= ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5836_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H2 @ B5 ) )
= ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5837_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H2 @ B5 ) )
= ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5838_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H2 @ B5 ) )
= ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5839_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
= ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5840_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H2 @ B5 ) )
= ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5841_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
= ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5842_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H2 @ B5 ) )
= ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5843_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H2 @ B5 ) )
= ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5844_sum_Osame__carrier,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ A4 )
= ( groups2073611262835488442omplex @ H2 @ B5 ) )
= ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5845_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5846_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5847_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5848_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5849_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5850_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5851_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5852_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5853_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5854_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5855_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5856_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5857_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5858_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5859_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5860_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5861_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5862_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5863_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5864_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5865_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5866_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5867_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5868_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5869_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5870_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5871_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5872_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5873_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5874_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5875_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5876_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5877_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5878_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5879_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B: nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5880_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5881_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5882_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5883_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5884_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B: nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5885_mask__numeral,axiom,
! [N: num] :
( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5886_mask__numeral,axiom,
! [N: num] :
( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5887_set__encode__insert,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_5888_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_5889_neg__numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_5890_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_5891_ceiling__less__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_5892_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_5893_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_5894_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_5895_singletonI,axiom,
! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singletonI
thf(fact_5896_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_5897_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_5898_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_5899_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_5900_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_5901_finite__insert,axiom,
! [A: real,A4: set_real] :
( ( finite_finite_real @ ( insert_real @ A @ A4 ) )
= ( finite_finite_real @ A4 ) ) ).
% finite_insert
thf(fact_5902_finite__insert,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) )
= ( finite4343798906461161616at_nat @ A4 ) ) ).
% finite_insert
thf(fact_5903_finite__insert,axiom,
! [A: nat,A4: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ).
% finite_insert
thf(fact_5904_finite__insert,axiom,
! [A: int,A4: set_int] :
( ( finite_finite_int @ ( insert_int @ A @ A4 ) )
= ( finite_finite_int @ A4 ) ) ).
% finite_insert
thf(fact_5905_finite__insert,axiom,
! [A: complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) )
= ( finite3207457112153483333omplex @ A4 ) ) ).
% finite_insert
thf(fact_5906_finite__insert,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) )
= ( finite6177210948735845034at_nat @ A4 ) ) ).
% finite_insert
thf(fact_5907_singleton__conv,axiom,
! [A: produc3843707927480180839at_nat] :
( ( collec6321179662152712658at_nat
@ ^ [X5: produc3843707927480180839at_nat] : X5 = A )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singleton_conv
thf(fact_5908_singleton__conv,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ^ [X5: list_nat] : X5 = A )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv
thf(fact_5909_singleton__conv,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ^ [X5: set_nat] : X5 = A )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_5910_singleton__conv,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] : X5 = A )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv
thf(fact_5911_singleton__conv,axiom,
! [A: real] :
( ( collect_real
@ ^ [X5: real] : X5 = A )
= ( insert_real @ A @ bot_bot_set_real ) ) ).
% singleton_conv
thf(fact_5912_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X5: nat] : X5 = A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_5913_singleton__conv,axiom,
! [A: int] :
( ( collect_int
@ ^ [X5: int] : X5 = A )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% singleton_conv
thf(fact_5914_singleton__conv2,axiom,
! [A: produc3843707927480180839at_nat] :
( ( collec6321179662152712658at_nat
@ ( ^ [Y4: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] : Y4 = Z2
@ A ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singleton_conv2
thf(fact_5915_singleton__conv2,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2
@ A ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv2
thf(fact_5916_singleton__conv2,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2
@ A ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_5917_singleton__conv2,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ( ^ [Y4: product_prod_nat_nat,Z2: product_prod_nat_nat] : Y4 = Z2
@ A ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv2
thf(fact_5918_singleton__conv2,axiom,
! [A: real] :
( ( collect_real
@ ( ^ [Y4: real,Z2: real] : Y4 = Z2
@ A ) )
= ( insert_real @ A @ bot_bot_set_real ) ) ).
% singleton_conv2
thf(fact_5919_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z2: nat] : Y4 = Z2
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_5920_singleton__conv2,axiom,
! [A: int] :
( ( collect_int
@ ( ^ [Y4: int,Z2: int] : Y4 = Z2
@ A ) )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% singleton_conv2
thf(fact_5921_singleton__insert__inj__eq_H,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ A4 )
= ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
= ( ( A = B )
& ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5922_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ A4 )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( ( A = B )
& ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5923_singleton__insert__inj__eq_H,axiom,
! [A: real,A4: set_real,B: real] :
( ( ( insert_real @ A @ A4 )
= ( insert_real @ B @ bot_bot_set_real ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A4 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5924_singleton__insert__inj__eq_H,axiom,
! [A: nat,A4: set_nat,B: nat] :
( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5925_singleton__insert__inj__eq_H,axiom,
! [A: int,A4: set_int,B: int] :
( ( ( insert_int @ A @ A4 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5926_singleton__insert__inj__eq,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
= ( insert9069300056098147895at_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5927_singleton__insert__inj__eq,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5928_singleton__insert__inj__eq,axiom,
! [B: real,A: real,A4: set_real] :
( ( ( insert_real @ B @ bot_bot_set_real )
= ( insert_real @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A4 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5929_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A4: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5930_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A4: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5931_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_5932_atLeastAtMost__singleton,axiom,
! [A: int] :
( ( set_or1266510415728281911st_int @ A @ A )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% atLeastAtMost_singleton
thf(fact_5933_atLeastAtMost__singleton,axiom,
! [A: real] :
( ( set_or1222579329274155063t_real @ A @ A )
= ( insert_real @ A @ bot_bot_set_real ) ) ).
% atLeastAtMost_singleton
thf(fact_5934_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C2 @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C2 ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_5935_atLeastAtMost__singleton__iff,axiom,
! [A: int,B: int,C2: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= ( insert_int @ C2 @ bot_bot_set_int ) )
= ( ( A = B )
& ( B = C2 ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_5936_atLeastAtMost__singleton__iff,axiom,
! [A: real,B: real,C2: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= ( insert_real @ C2 @ bot_bot_set_real ) )
= ( ( A = B )
& ( B = C2 ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_5937_insert__Diff__single,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( insert9069300056098147895at_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_5938_insert__Diff__single,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( insert8211810215607154385at_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_5939_insert__Diff__single,axiom,
! [A: real,A4: set_real] :
( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( insert_real @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_5940_insert__Diff__single,axiom,
! [A: int,A4: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_5941_insert__Diff__single,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_5942_finite__Diff__insert,axiom,
! [A4: set_real,A: real,B5: set_real] :
( ( finite_finite_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B5 ) ) )
= ( finite_finite_real @ ( minus_minus_set_real @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5943_finite__Diff__insert,axiom,
! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) ) )
= ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5944_finite__Diff__insert,axiom,
! [A4: set_int,A: int,B5: set_int] :
( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B5 ) ) )
= ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5945_finite__Diff__insert,axiom,
! [A4: set_complex,A: complex,B5: set_complex] :
( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ B5 ) ) )
= ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5946_finite__Diff__insert,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) ) )
= ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5947_finite__Diff__insert,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_5948_ceiling__zero,axiom,
( ( archim2889992004027027881ng_rat @ zero_zero_rat )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_5949_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_5950_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_5951_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_5952_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_5953_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_5954_sum_Oinsert,axiom,
! [A4: set_real,X: real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5955_sum_Oinsert,axiom,
! [A4: set_int,X: int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ~ ( member_int @ X @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5956_sum_Oinsert,axiom,
! [A4: set_complex,X: complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5957_sum_Oinsert,axiom,
! [A4: set_real,X: real,G: real > int] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5958_sum_Oinsert,axiom,
! [A4: set_nat,X: nat,G: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5959_sum_Oinsert,axiom,
! [A4: set_complex,X: complex,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5960_sum_Oinsert,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5961_sum_Oinsert,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ~ ( member_int @ X @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5962_sum_Oinsert,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5963_sum_Oinsert,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% sum.insert
thf(fact_5964_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_5965_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_5966_subset__Compl__singleton,axiom,
! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
= ( ~ ( member8757157785044589968at_nat @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5967_subset__Compl__singleton,axiom,
! [A4: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5968_subset__Compl__singleton,axiom,
! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
= ( ~ ( member8440522571783428010at_nat @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5969_subset__Compl__singleton,axiom,
! [A4: set_real,B: real] :
( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( ~ ( member_real @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5970_subset__Compl__singleton,axiom,
! [A4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5971_subset__Compl__singleton,axiom,
! [A4: set_int,B: int] :
( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_5972_ceiling__add__of__int,axiom,
! [X: rat,Z: int] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% ceiling_add_of_int
thf(fact_5973_ceiling__add__of__int,axiom,
! [X: real,Z: int] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% ceiling_add_of_int
thf(fact_5974_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_5975_ceiling__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_le_zero
thf(fact_5976_zero__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% zero_less_ceiling
thf(fact_5977_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_5978_ceiling__le__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_5979_ceiling__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_5980_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_5981_ceiling__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_less_one
thf(fact_5982_one__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% one_le_ceiling
thf(fact_5983_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_5984_numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_5985_numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_5986_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_5987_ceiling__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% ceiling_le_one
thf(fact_5988_one__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ one_one_rat @ X ) ) ).
% one_less_ceiling
thf(fact_5989_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_5990_set__replicate,axiom,
! [N: nat,X: produc3843707927480180839at_nat] :
( ( N != zero_zero_nat )
=> ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% set_replicate
thf(fact_5991_set__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( N != zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% set_replicate
thf(fact_5992_set__replicate,axiom,
! [N: nat,X: product_prod_nat_nat] :
( ( N != zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% set_replicate
thf(fact_5993_set__replicate,axiom,
! [N: nat,X: real] :
( ( N != zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X ) )
= ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% set_replicate
thf(fact_5994_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_5995_set__replicate,axiom,
! [N: nat,X: int] :
( ( N != zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% set_replicate
thf(fact_5996_ceiling__add__numeral,axiom,
! [X: rat,V: num] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_add_numeral
thf(fact_5997_ceiling__add__numeral,axiom,
! [X: real,V: num] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_add_numeral
thf(fact_5998_ceiling__add__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_5999_ceiling__add__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_6000_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6001_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6002_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6003_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6004_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6005_ceiling__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% ceiling_less_zero
thf(fact_6006_ceiling__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% ceiling_less_zero
thf(fact_6007_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_6008_zero__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% zero_le_ceiling
thf(fact_6009_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_6010_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_6011_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_6012_ceiling__less__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% ceiling_less_numeral
thf(fact_6013_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% ceiling_less_numeral
thf(fact_6014_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_6015_numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_6016_ceiling__le__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_6017_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_6018_neg__numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_6019_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_6020_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= ( divide1717551699836669952omplex @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_6021_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > rat,D: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= ( divide_divide_rat @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_6022_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= ( divide_divide_real @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_6023_singletonD,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6024_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6025_singletonD,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6026_singletonD,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6027_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6028_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6029_singleton__iff,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6030_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6031_singleton__iff,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6032_singleton__iff,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6033_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6034_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6035_doubleton__eq__iff,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
= ( insert9069300056098147895at_nat @ C2 @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6036_doubleton__eq__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C2: product_prod_nat_nat,D: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ C2 @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6037_doubleton__eq__iff,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
= ( insert_real @ C2 @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6038_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6039_doubleton__eq__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C2 @ ( insert_int @ D @ bot_bot_set_int ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6040_insert__not__empty,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( insert9069300056098147895at_nat @ A @ A4 )
!= bot_bo228742789529271731at_nat ) ).
% insert_not_empty
thf(fact_6041_insert__not__empty,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ A4 )
!= bot_bo2099793752762293965at_nat ) ).
% insert_not_empty
thf(fact_6042_insert__not__empty,axiom,
! [A: real,A4: set_real] :
( ( insert_real @ A @ A4 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_6043_insert__not__empty,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_6044_insert__not__empty,axiom,
! [A: int,A4: set_int] :
( ( insert_int @ A @ A4 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_6045_singleton__inject,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
= ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6046_singleton__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6047_singleton__inject,axiom,
! [A: real,B: real] :
( ( ( insert_real @ A @ bot_bot_set_real )
= ( insert_real @ B @ bot_bot_set_real ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6048_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6049_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int @ A @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6050_finite_OinsertI,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( finite_finite_real @ ( insert_real @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6051_finite_OinsertI,axiom,
! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ( finite4343798906461161616at_nat @ A4 )
=> ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6052_finite_OinsertI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6053_finite_OinsertI,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( finite_finite_int @ ( insert_int @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6054_finite_OinsertI,axiom,
! [A4: set_complex,A: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6055_finite_OinsertI,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_6056_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_6057_sum__diff1__nat,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
( ( ( member8757157785044589968at_nat @ A @ A4 )
=> ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member8757157785044589968at_nat @ A @ A4 )
=> ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( groups3860910324918113789at_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6058_sum__diff1__nat,axiom,
! [A: set_nat,A4: set_set_nat,F: set_nat > nat] :
( ( ( member_set_nat @ A @ A4 )
=> ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_set_nat @ A @ A4 )
=> ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( groups8294997508430121362at_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6059_sum__diff1__nat,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( groups977919841031483927at_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6060_sum__diff1__nat,axiom,
! [A: real,A4: set_real,F: real > nat] :
( ( ( member_real @ A @ A4 )
=> ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6061_sum__diff1__nat,axiom,
! [A: int,A4: set_int,F: int > nat] :
( ( ( member_int @ A @ A4 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A4 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6062_sum__diff1__nat,axiom,
! [A: nat,A4: set_nat,F: nat > nat] :
( ( ( member_nat @ A @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_6063_Collect__conv__if,axiom,
! [P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
( ( ( P2 @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X5: produc3843707927480180839at_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X5: produc3843707927480180839at_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bo228742789529271731at_nat ) ) ) ).
% Collect_conv_if
thf(fact_6064_Collect__conv__if,axiom,
! [P2: list_nat > $o,A: list_nat] :
( ( ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X5: list_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X5: list_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if
thf(fact_6065_Collect__conv__if,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ( ( P2 @ A )
=> ( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_6066_Collect__conv__if,axiom,
! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P2 @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if
thf(fact_6067_Collect__conv__if,axiom,
! [P2: real > $o,A: real] :
( ( ( P2 @ A )
=> ( ( collect_real
@ ^ [X5: real] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert_real @ A @ bot_bot_set_real ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_real
@ ^ [X5: real] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bot_set_real ) ) ) ).
% Collect_conv_if
thf(fact_6068_Collect__conv__if,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_6069_Collect__conv__if,axiom,
! [P2: int > $o,A: int] :
( ( ( P2 @ A )
=> ( ( collect_int
@ ^ [X5: int] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= ( insert_int @ A @ bot_bot_set_int ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_int
@ ^ [X5: int] :
( ( X5 = A )
& ( P2 @ X5 ) ) )
= bot_bot_set_int ) ) ) ).
% Collect_conv_if
thf(fact_6070_Collect__conv__if2,axiom,
! [P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
( ( ( P2 @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X5: produc3843707927480180839at_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X5: produc3843707927480180839at_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bo228742789529271731at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6071_Collect__conv__if2,axiom,
! [P2: list_nat > $o,A: list_nat] :
( ( ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X5: list_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X5: list_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6072_Collect__conv__if2,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ( ( P2 @ A )
=> ( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6073_Collect__conv__if2,axiom,
! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P2 @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6074_Collect__conv__if2,axiom,
! [P2: real > $o,A: real] :
( ( ( P2 @ A )
=> ( ( collect_real
@ ^ [X5: real] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert_real @ A @ bot_bot_set_real ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_real
@ ^ [X5: real] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bot_set_real ) ) ) ).
% Collect_conv_if2
thf(fact_6075_Collect__conv__if2,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6076_Collect__conv__if2,axiom,
! [P2: int > $o,A: int] :
( ( ( P2 @ A )
=> ( ( collect_int
@ ^ [X5: int] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= ( insert_int @ A @ bot_bot_set_int ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_int
@ ^ [X5: int] :
( ( A = X5 )
& ( P2 @ X5 ) ) )
= bot_bot_set_int ) ) ) ).
% Collect_conv_if2
thf(fact_6077_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6078_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A4 )
= ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6079_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_6080_ceiling__mono,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% ceiling_mono
thf(fact_6081_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_6082_le__of__int__ceiling,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_6083_ceiling__less__cancel,axiom,
! [X: rat,Y: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_6084_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_6085_infinite__finite__induct,axiom,
! [P2: set_Pr4329608150637261639at_nat > $o,A4: set_Pr4329608150637261639at_nat] :
( ! [A6: set_Pr4329608150637261639at_nat] :
( ~ ( finite4343798906461161616at_nat @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6086_infinite__finite__induct,axiom,
! [P2: set_set_nat > $o,A4: set_set_nat] :
( ! [A6: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X4: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ~ ( member_set_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6087_infinite__finite__induct,axiom,
! [P2: set_complex > $o,A4: set_complex] :
( ! [A6: set_complex] :
( ~ ( finite3207457112153483333omplex @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [X4: complex,F4: set_complex] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ~ ( member_complex @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_complex @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6088_infinite__finite__induct,axiom,
! [P2: set_Pr1261947904930325089at_nat > $o,A4: set_Pr1261947904930325089at_nat] :
( ! [A6: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6089_infinite__finite__induct,axiom,
! [P2: set_real > $o,A4: set_real] :
( ! [A6: set_real] :
( ~ ( finite_finite_real @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [X4: real,F4: set_real] :
( ( finite_finite_real @ F4 )
=> ( ~ ( member_real @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_real @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6090_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A4: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6091_infinite__finite__induct,axiom,
! [P2: set_int > $o,A4: set_int] :
( ! [A6: set_int] :
( ~ ( finite_finite_int @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [X4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_int @ X4 @ F4 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_6092_finite__ne__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F3 )
=> ( ( F3 != bot_bo228742789529271731at_nat )
=> ( ! [X4: produc3843707927480180839at_nat] : ( P2 @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
=> ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( F4 != bot_bo228742789529271731at_nat )
=> ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6093_finite__ne__induct,axiom,
! [F3: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( F3 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] : ( P2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
=> ( ! [X4: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( F4 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6094_finite__ne__induct,axiom,
! [F3: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( F3 != bot_bot_set_complex )
=> ( ! [X4: complex] : ( P2 @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
=> ( ! [X4: complex,F4: set_complex] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( F4 != bot_bot_set_complex )
=> ( ~ ( member_complex @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6095_finite__ne__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( F3 != bot_bo2099793752762293965at_nat )
=> ( ! [X4: product_prod_nat_nat] : ( P2 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
=> ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( F4 != bot_bo2099793752762293965at_nat )
=> ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6096_finite__ne__induct,axiom,
! [F3: set_real,P2: set_real > $o] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ! [X4: real] : ( P2 @ ( insert_real @ X4 @ bot_bot_set_real ) )
=> ( ! [X4: real,F4: set_real] :
( ( finite_finite_real @ F4 )
=> ( ( F4 != bot_bot_set_real )
=> ( ~ ( member_real @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6097_finite__ne__induct,axiom,
! [F3: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X4: nat] : ( P2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ! [X4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6098_finite__ne__induct,axiom,
! [F3: set_int,P2: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( F3 != bot_bot_set_int )
=> ( ! [X4: int] : ( P2 @ ( insert_int @ X4 @ bot_bot_set_int ) )
=> ( ! [X4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( F4 != bot_bot_set_int )
=> ( ~ ( member_int @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_6099_finite__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F3 )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6100_finite__induct,axiom,
! [F3: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X4: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ~ ( member_set_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6101_finite__induct,axiom,
! [F3: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [X4: complex,F4: set_complex] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ~ ( member_complex @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_complex @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6102_finite__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6103_finite__induct,axiom,
! [F3: set_real,P2: set_real > $o] :
( ( finite_finite_real @ F3 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [X4: real,F4: set_real] :
( ( finite_finite_real @ F4 )
=> ( ~ ( member_real @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_real @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6104_finite__induct,axiom,
! [F3: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6105_finite__induct,axiom,
! [F3: set_int,P2: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [X4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_int @ X4 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_6106_finite_Osimps,axiom,
( finite4343798906461161616at_nat
= ( ^ [A5: set_Pr4329608150637261639at_nat] :
( ( A5 = bot_bo228742789529271731at_nat )
| ? [A7: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat] :
( ( A5
= ( insert9069300056098147895at_nat @ B4 @ A7 ) )
& ( finite4343798906461161616at_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6107_finite_Osimps,axiom,
( finite3207457112153483333omplex
= ( ^ [A5: set_complex] :
( ( A5 = bot_bot_set_complex )
| ? [A7: set_complex,B4: complex] :
( ( A5
= ( insert_complex @ B4 @ A7 ) )
& ( finite3207457112153483333omplex @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6108_finite_Osimps,axiom,
( finite6177210948735845034at_nat
= ( ^ [A5: set_Pr1261947904930325089at_nat] :
( ( A5 = bot_bo2099793752762293965at_nat )
| ? [A7: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
( ( A5
= ( insert8211810215607154385at_nat @ B4 @ A7 ) )
& ( finite6177210948735845034at_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6109_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A5: set_real] :
( ( A5 = bot_bot_set_real )
| ? [A7: set_real,B4: real] :
( ( A5
= ( insert_real @ B4 @ A7 ) )
& ( finite_finite_real @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6110_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A7: set_nat,B4: nat] :
( ( A5
= ( insert_nat @ B4 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6111_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A5: set_int] :
( ( A5 = bot_bot_set_int )
| ? [A7: set_int,B4: int] :
( ( A5
= ( insert_int @ B4 @ A7 ) )
& ( finite_finite_int @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_6112_finite_Ocases,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A )
=> ( ( A != bot_bo228742789529271731at_nat )
=> ~ ! [A6: set_Pr4329608150637261639at_nat] :
( ? [A3: produc3843707927480180839at_nat] :
( A
= ( insert9069300056098147895at_nat @ A3 @ A6 ) )
=> ~ ( finite4343798906461161616at_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6113_finite_Ocases,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( A != bot_bot_set_complex )
=> ~ ! [A6: set_complex] :
( ? [A3: complex] :
( A
= ( insert_complex @ A3 @ A6 ) )
=> ~ ( finite3207457112153483333omplex @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6114_finite_Ocases,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( A != bot_bo2099793752762293965at_nat )
=> ~ ! [A6: set_Pr1261947904930325089at_nat] :
( ? [A3: product_prod_nat_nat] :
( A
= ( insert8211810215607154385at_nat @ A3 @ A6 ) )
=> ~ ( finite6177210948735845034at_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6115_finite_Ocases,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ~ ! [A6: set_real] :
( ? [A3: real] :
( A
= ( insert_real @ A3 @ A6 ) )
=> ~ ( finite_finite_real @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6116_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6117_finite_Ocases,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ~ ! [A6: set_int] :
( ? [A3: int] :
( A
= ( insert_int @ A3 @ A6 ) )
=> ~ ( finite_finite_int @ A6 ) ) ) ) ).
% finite.cases
thf(fact_6118_subset__singleton__iff,axiom,
! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
= ( ( X7 = bot_bo228742789529271731at_nat )
| ( X7
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6119_subset__singleton__iff,axiom,
! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( ( X7 = bot_bo2099793752762293965at_nat )
| ( X7
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6120_subset__singleton__iff,axiom,
! [X7: set_real,A: real] :
( ( ord_less_eq_set_real @ X7 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( ( X7 = bot_bot_set_real )
| ( X7
= ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_6121_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6122_subset__singleton__iff,axiom,
! [X7: set_int,A: int] :
( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
= ( ( X7 = bot_bot_set_int )
| ( X7
= ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_6123_subset__singletonD,axiom,
! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
=> ( ( A4 = bot_bo228742789529271731at_nat )
| ( A4
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6124_subset__singletonD,axiom,
! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
=> ( ( A4 = bot_bo2099793752762293965at_nat )
| ( A4
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6125_subset__singletonD,axiom,
! [A4: set_real,X: real] :
( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) )
=> ( ( A4 = bot_bot_set_real )
| ( A4
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_6126_subset__singletonD,axiom,
! [A4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A4 = bot_bot_set_nat )
| ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6127_subset__singletonD,axiom,
! [A4: set_int,X: int] :
( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( A4 = bot_bot_set_int )
| ( A4
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_6128_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_6129_atLeastAtMost__singleton_H,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_6130_atLeastAtMost__singleton_H,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_6131_Diff__insert__absorb,axiom,
! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ~ ( member8757157785044589968at_nat @ X @ A4 )
=> ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6132_Diff__insert__absorb,axiom,
! [X: set_nat,A4: set_set_nat] :
( ~ ( member_set_nat @ X @ A4 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6133_Diff__insert__absorb,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6134_Diff__insert__absorb,axiom,
! [X: real,A4: set_real] :
( ~ ( member_real @ X @ A4 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A4 ) @ ( insert_real @ X @ bot_bot_set_real ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6135_Diff__insert__absorb,axiom,
! [X: int,A4: set_int] :
( ~ ( member_int @ X @ A4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6136_Diff__insert__absorb,axiom,
! [X: nat,A4: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_6137_Diff__insert2,axiom,
! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
= ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_6138_Diff__insert2,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_6139_Diff__insert2,axiom,
! [A4: set_real,A: real,B5: set_real] :
( ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B5 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_6140_Diff__insert2,axiom,
! [A4: set_int,A: int,B5: set_int] :
( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B5 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_6141_Diff__insert2,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_6142_insert__Diff,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ A4 )
=> ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6143_insert__Diff,axiom,
! [A: set_nat,A4: set_set_nat] :
( ( member_set_nat @ A @ A4 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6144_insert__Diff,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6145_insert__Diff,axiom,
! [A: real,A4: set_real] :
( ( member_real @ A @ A4 )
=> ( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6146_insert__Diff,axiom,
! [A: int,A4: set_int] :
( ( member_int @ A @ A4 )
=> ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6147_insert__Diff,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_6148_Diff__insert,axiom,
! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
= ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% Diff_insert
thf(fact_6149_Diff__insert,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% Diff_insert
thf(fact_6150_Diff__insert,axiom,
! [A4: set_real,A: real,B5: set_real] :
( ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B5 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A4 @ B5 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_6151_Diff__insert,axiom,
! [A4: set_int,A: int,B5: set_int] :
( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B5 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_6152_Diff__insert,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_6153_sum__subtractf__nat,axiom,
! [A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups977919841031483927at_nat
@ ^ [X5: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6154_sum__subtractf__nat,axiom,
! [A4: set_real,G: real > nat,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X5: real] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6155_sum__subtractf__nat,axiom,
! [A4: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X5: set_nat] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( groups8294997508430121362at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6156_sum__subtractf__nat,axiom,
! [A4: set_int,G: int > nat,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6157_sum__subtractf__nat,axiom,
! [A4: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6158_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6159_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6160_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_6161_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6162_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M2: nat,K2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6163_sum__eq__Suc0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: int] :
( ( member_int @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6164_sum__eq__Suc0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: complex] :
( ( member_complex @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6165_sum__eq__Suc0__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups977919841031483927at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6166_sum__eq__Suc0__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6167_sum__SucD,axiom,
! [F: nat > nat,A4: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_6168_sum__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: int] :
( ( member_int @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6169_sum__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: complex] :
( ( member_complex @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6170_sum__eq__1__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups977919841031483927at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6171_sum__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6172_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% ceiling_le
thf(fact_6173_ceiling__le,axiom,
! [X: rat,A: int] :
( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% ceiling_le
thf(fact_6174_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_6175_ceiling__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_6176_less__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_6177_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_6178_ceiling__add__le,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% ceiling_add_le
thf(fact_6179_ceiling__add__le,axiom,
! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% ceiling_add_le
thf(fact_6180_finite__ranking__induct,axiom,
! [S2: set_complex,P2: set_complex > $o,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [X4: complex,S6: set_complex] :
( ( finite3207457112153483333omplex @ S6 )
=> ( ! [Y6: complex] :
( ( member_complex @ Y6 @ S6 )
=> ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_complex @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6181_finite__ranking__induct,axiom,
! [S2: set_real,P2: set_real > $o,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [X4: real,S6: set_real] :
( ( finite_finite_real @ S6 )
=> ( ! [Y6: real] :
( ( member_real @ Y6 @ S6 )
=> ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_real @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6182_finite__ranking__induct,axiom,
! [S2: set_nat,P2: set_nat > $o,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X4: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S6 )
=> ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_nat @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6183_finite__ranking__induct,axiom,
! [S2: set_int,P2: set_int > $o,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [X4: int,S6: set_int] :
( ( finite_finite_int @ S6 )
=> ( ! [Y6: int] :
( ( member_int @ Y6 @ S6 )
=> ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_int @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6184_finite__ranking__induct,axiom,
! [S2: set_complex,P2: set_complex > $o,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [X4: complex,S6: set_complex] :
( ( finite3207457112153483333omplex @ S6 )
=> ( ! [Y6: complex] :
( ( member_complex @ Y6 @ S6 )
=> ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_complex @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6185_finite__ranking__induct,axiom,
! [S2: set_real,P2: set_real > $o,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [X4: real,S6: set_real] :
( ( finite_finite_real @ S6 )
=> ( ! [Y6: real] :
( ( member_real @ Y6 @ S6 )
=> ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_real @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6186_finite__ranking__induct,axiom,
! [S2: set_nat,P2: set_nat > $o,F: nat > num] :
( ( finite_finite_nat @ S2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X4: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S6 )
=> ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_nat @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6187_finite__ranking__induct,axiom,
! [S2: set_int,P2: set_int > $o,F: int > num] :
( ( finite_finite_int @ S2 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [X4: int,S6: set_int] :
( ( finite_finite_int @ S6 )
=> ( ! [Y6: int] :
( ( member_int @ Y6 @ S6 )
=> ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_int @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6188_finite__ranking__induct,axiom,
! [S2: set_complex,P2: set_complex > $o,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [X4: complex,S6: set_complex] :
( ( finite3207457112153483333omplex @ S6 )
=> ( ! [Y6: complex] :
( ( member_complex @ Y6 @ S6 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_complex @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6189_finite__ranking__induct,axiom,
! [S2: set_real,P2: set_real > $o,F: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [X4: real,S6: set_real] :
( ( finite_finite_real @ S6 )
=> ( ! [Y6: real] :
( ( member_real @ Y6 @ S6 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert_real @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_6190_finite__linorder__max__induct,axiom,
! [A4: set_real,P2: set_real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [B3: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( ord_less_real @ X3 @ B3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_real @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_6191_finite__linorder__max__induct,axiom,
! [A4: set_rat,P2: set_rat > $o] :
( ( finite_finite_rat @ A4 )
=> ( ( P2 @ bot_bot_set_rat )
=> ( ! [B3: rat,A6: set_rat] :
( ( finite_finite_rat @ A6 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( ord_less_rat @ X3 @ B3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_rat @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_6192_finite__linorder__max__induct,axiom,
! [A4: set_num,P2: set_num > $o] :
( ( finite_finite_num @ A4 )
=> ( ( P2 @ bot_bot_set_num )
=> ( ! [B3: num,A6: set_num] :
( ( finite_finite_num @ A6 )
=> ( ! [X3: num] :
( ( member_num @ X3 @ A6 )
=> ( ord_less_num @ X3 @ B3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_num @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_6193_finite__linorder__max__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ord_less_nat @ X3 @ B3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_nat @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_6194_finite__linorder__max__induct,axiom,
! [A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [B3: int,A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( ord_less_int @ X3 @ B3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_int @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_6195_finite__linorder__min__induct,axiom,
! [A4: set_real,P2: set_real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [B3: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( ord_less_real @ B3 @ X3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_real @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_6196_finite__linorder__min__induct,axiom,
! [A4: set_rat,P2: set_rat > $o] :
( ( finite_finite_rat @ A4 )
=> ( ( P2 @ bot_bot_set_rat )
=> ( ! [B3: rat,A6: set_rat] :
( ( finite_finite_rat @ A6 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( ord_less_rat @ B3 @ X3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_rat @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_6197_finite__linorder__min__induct,axiom,
! [A4: set_num,P2: set_num > $o] :
( ( finite_finite_num @ A4 )
=> ( ( P2 @ bot_bot_set_num )
=> ( ! [B3: num,A6: set_num] :
( ( finite_finite_num @ A6 )
=> ( ! [X3: num] :
( ( member_num @ X3 @ A6 )
=> ( ord_less_num @ B3 @ X3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_num @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_6198_finite__linorder__min__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ord_less_nat @ B3 @ X3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_nat @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_6199_finite__linorder__min__induct,axiom,
! [A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [B3: int,A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( ord_less_int @ B3 @ X3 ) )
=> ( ( P2 @ A6 )
=> ( P2 @ ( insert_int @ B3 @ A6 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_6200_sum_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A4 ) )
= ( groups1935376822645274424al_nat @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6201_sum_Oinsert__if,axiom,
! [A4: set_int,X: int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ X @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
= ( groups4541462559716669496nt_nat @ G @ A4 ) ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6202_sum_Oinsert__if,axiom,
! [A4: set_complex,X: complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ X @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A4 ) )
= ( groups5693394587270226106ex_nat @ G @ A4 ) ) )
& ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6203_sum_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A4 ) )
= ( groups1932886352136224148al_int @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6204_sum_Oinsert__if,axiom,
! [A4: set_nat,X: nat,G: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
= ( groups3539618377306564664at_int @ G @ A4 ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6205_sum_Oinsert__if,axiom,
! [A4: set_complex,X: complex,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ X @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A4 ) )
= ( groups5690904116761175830ex_int @ G @ A4 ) ) )
& ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6206_sum_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
= ( groups8097168146408367636l_real @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6207_sum_Oinsert__if,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ X @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
= ( groups8778361861064173332t_real @ G @ A4 ) ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6208_sum_Oinsert__if,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ X @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( groups5808333547571424918x_real @ G @ A4 ) ) )
& ( ~ ( member_complex @ X @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6209_sum_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( groups1300246762558778688al_rat @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6210_finite__subset__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F3 )
=> ( ( ord_le1268244103169919719at_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( member8757157785044589968at_nat @ A3 @ A4 )
=> ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6211_finite__subset__induct,axiom,
! [F3: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A3: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( member_set_nat @ A3 @ A4 )
=> ( ~ ( member_set_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6212_finite__subset__induct,axiom,
! [F3: set_complex,A4: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( ord_le211207098394363844omplex @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [A3: complex,F4: set_complex] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( ~ ( member_complex @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_complex @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6213_finite__subset__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( ord_le3146513528884898305at_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( member8440522571783428010at_nat @ A3 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6214_finite__subset__induct,axiom,
! [F3: set_real,A4: set_real,P2: set_real > $o] :
( ( finite_finite_real @ F3 )
=> ( ( ord_less_eq_set_real @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [A3: real,F4: set_real] :
( ( finite_finite_real @ F4 )
=> ( ( member_real @ A3 @ A4 )
=> ( ~ ( member_real @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_real @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6215_finite__subset__induct,axiom,
! [F3: set_nat,A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6216_finite__subset__induct,axiom,
! [F3: set_int,A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [A3: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A3 @ A4 )
=> ( ~ ( member_int @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_int @ A3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_6217_finite__subset__induct_H,axiom,
! [F3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F3 )
=> ( ( ord_le1268244103169919719at_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( member8757157785044589968at_nat @ A3 @ A4 )
=> ( ( ord_le1268244103169919719at_nat @ F4 @ A4 )
=> ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6218_finite__subset__induct_H,axiom,
! [F3: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A3: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( member_set_nat @ A3 @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ F4 @ A4 )
=> ( ~ ( member_set_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6219_finite__subset__induct_H,axiom,
! [F3: set_complex,A4: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ F3 )
=> ( ( ord_le211207098394363844omplex @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [A3: complex,F4: set_complex] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( member_complex @ A3 @ A4 )
=> ( ( ord_le211207098394363844omplex @ F4 @ A4 )
=> ( ~ ( member_complex @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_complex @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6220_finite__subset__induct_H,axiom,
! [F3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( ( ord_le3146513528884898305at_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( member8440522571783428010at_nat @ A3 @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ F4 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6221_finite__subset__induct_H,axiom,
! [F3: set_real,A4: set_real,P2: set_real > $o] :
( ( finite_finite_real @ F3 )
=> ( ( ord_less_eq_set_real @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [A3: real,F4: set_real] :
( ( finite_finite_real @ F4 )
=> ( ( member_real @ A3 @ A4 )
=> ( ( ord_less_eq_set_real @ F4 @ A4 )
=> ( ~ ( member_real @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_real @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6222_finite__subset__induct_H,axiom,
! [F3: set_nat,A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A4 )
=> ( ~ ( member_nat @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6223_finite__subset__induct_H,axiom,
! [F3: set_int,A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A4 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [A3: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A3 @ A4 )
=> ( ( ord_less_eq_set_int @ F4 @ A4 )
=> ( ~ ( member_int @ A3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_int @ A3 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_6224_finite__empty__induct,axiom,
! [A4: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: produc3843707927480180839at_nat,A6: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A6 )
=> ( ( member8757157785044589968at_nat @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
=> ( P2 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_6225_finite__empty__induct,axiom,
! [A4: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: set_nat,A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( member_set_nat @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_6226_finite__empty__induct,axiom,
! [A4: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: complex,A6: set_complex] :
( ( finite3207457112153483333omplex @ A6 )
=> ( ( member_complex @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
=> ( P2 @ bot_bot_set_complex ) ) ) ) ).
% finite_empty_induct
thf(fact_6227_finite__empty__induct,axiom,
! [A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A6 )
=> ( ( member8440522571783428010at_nat @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ( P2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_6228_finite__empty__induct,axiom,
! [A4: set_real,P2: set_real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( member_real @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_real @ A6 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
=> ( P2 @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_6229_finite__empty__induct,axiom,
! [A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: int,A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ( member_int @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_int @ A6 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
=> ( P2 @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_6230_finite__empty__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A3 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_6231_infinite__coinduct,axiom,
! [X7: set_Pr4329608150637261639at_nat > $o,A4: set_Pr4329608150637261639at_nat] :
( ( X7 @ A4 )
=> ( ! [A6: set_Pr4329608150637261639at_nat] :
( ( X7 @ A6 )
=> ? [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A6 )
& ( ( X7 @ ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) )
| ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
=> ~ ( finite4343798906461161616at_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6232_infinite__coinduct,axiom,
! [X7: set_complex > $o,A4: set_complex] :
( ( X7 @ A4 )
=> ( ! [A6: set_complex] :
( ( X7 @ A6 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A6 )
& ( ( X7 @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) )
| ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) )
=> ~ ( finite3207457112153483333omplex @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6233_infinite__coinduct,axiom,
! [X7: set_Pr1261947904930325089at_nat > $o,A4: set_Pr1261947904930325089at_nat] :
( ( X7 @ A4 )
=> ( ! [A6: set_Pr1261947904930325089at_nat] :
( ( X7 @ A6 )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A6 )
& ( ( X7 @ ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) )
| ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ~ ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6234_infinite__coinduct,axiom,
! [X7: set_real > $o,A4: set_real] :
( ( X7 @ A4 )
=> ( ! [A6: set_real] :
( ( X7 @ A6 )
=> ? [X3: real] :
( ( member_real @ X3 @ A6 )
& ( ( X7 @ ( minus_minus_set_real @ A6 @ ( insert_real @ X3 @ bot_bot_set_real ) ) )
| ~ ( finite_finite_real @ ( minus_minus_set_real @ A6 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) )
=> ~ ( finite_finite_real @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6235_infinite__coinduct,axiom,
! [X7: set_int > $o,A4: set_int] :
( ( X7 @ A4 )
=> ( ! [A6: set_int] :
( ( X7 @ A6 )
=> ? [X3: int] :
( ( member_int @ X3 @ A6 )
& ( ( X7 @ ( minus_minus_set_int @ A6 @ ( insert_int @ X3 @ bot_bot_set_int ) ) )
| ~ ( finite_finite_int @ ( minus_minus_set_int @ A6 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) )
=> ~ ( finite_finite_int @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6236_infinite__coinduct,axiom,
! [X7: set_nat > $o,A4: set_nat] :
( ( X7 @ A4 )
=> ( ! [A6: set_nat] :
( ( X7 @ A6 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ( ( X7 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_6237_infinite__remove,axiom,
! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ~ ( finite4343798906461161616at_nat @ S2 )
=> ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% infinite_remove
thf(fact_6238_infinite__remove,axiom,
! [S2: set_complex,A: complex] :
( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% infinite_remove
thf(fact_6239_infinite__remove,axiom,
! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ~ ( finite6177210948735845034at_nat @ S2 )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% infinite_remove
thf(fact_6240_infinite__remove,axiom,
! [S2: set_real,A: real] :
( ~ ( finite_finite_real @ S2 )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% infinite_remove
thf(fact_6241_infinite__remove,axiom,
! [S2: set_int,A: int] :
( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% infinite_remove
thf(fact_6242_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_6243_subset__insert__iff,axiom,
! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
= ( ( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6244_subset__insert__iff,axiom,
! [A4: set_set_nat,X: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
= ( ( ( member_set_nat @ X @ A4 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
& ( ~ ( member_set_nat @ X @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6245_subset__insert__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
= ( ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6246_subset__insert__iff,axiom,
! [A4: set_real,X: real,B5: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X @ B5 ) )
= ( ( ( member_real @ X @ A4 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ord_less_eq_set_real @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6247_subset__insert__iff,axiom,
! [A4: set_nat,X: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
= ( ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6248_subset__insert__iff,axiom,
! [A4: set_int,X: int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B5 ) )
= ( ( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_6249_Diff__single__insert,axiom,
! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 )
=> ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_6250_Diff__single__insert,axiom,
! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 )
=> ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_6251_Diff__single__insert,axiom,
! [A4: set_real,X: real,B5: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 )
=> ( ord_less_eq_set_real @ A4 @ ( insert_real @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_6252_Diff__single__insert,axiom,
! [A4: set_nat,X: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_6253_Diff__single__insert,axiom,
! [A4: set_int,X: int,B5: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 )
=> ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_6254_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_6255_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_6256_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_6257_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or1269000886237332187st_nat @ M2 @ N )
= ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_6258_set__update__subset__insert,axiom,
! [Xs: list_P6011104703257516679at_nat,I2: nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I2 @ X ) ) @ ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6259_set__update__subset__insert,axiom,
! [Xs: list_real,I2: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I2 @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6260_set__update__subset__insert,axiom,
! [Xs: list_P5464809261938338413at_nat,I2: nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I2 @ X ) ) @ ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6261_set__update__subset__insert,axiom,
! [Xs: list_nat,I2: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6262_set__update__subset__insert,axiom,
! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6263_set__update__subset__insert,axiom,
! [Xs: list_int,I2: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I2 @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_6264_Compl__insert,axiom,
! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) )
= ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% Compl_insert
thf(fact_6265_Compl__insert,axiom,
! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) )
= ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% Compl_insert
thf(fact_6266_Compl__insert,axiom,
! [X: real,A4: set_real] :
( ( uminus612125837232591019t_real @ ( insert_real @ X @ A4 ) )
= ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A4 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% Compl_insert
thf(fact_6267_Compl__insert,axiom,
! [X: int,A4: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A4 ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_6268_Compl__insert,axiom,
! [X: nat,A4: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A4 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_6269_sum__power__add,axiom,
! [X: complex,M2: nat,I6: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M2 @ I4 ) )
@ I6 )
= ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_6270_sum__power__add,axiom,
! [X: rat,M2: nat,I6: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M2 @ I4 ) )
@ I6 )
= ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_6271_sum__power__add,axiom,
! [X: int,M2: nat,I6: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I4 ) )
@ I6 )
= ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_6272_sum__power__add,axiom,
! [X: real,M2: nat,I6: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M2 @ I4 ) )
@ I6 )
= ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_6273_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6274_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N: nat,M2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6275_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_6276_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_6277_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_6278_of__int__ceiling__le__add__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% of_int_ceiling_le_add_one
thf(fact_6279_of__int__ceiling__le__add__one,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% of_int_ceiling_le_add_one
thf(fact_6280_of__int__ceiling__diff__one__le,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_6281_of__int__ceiling__diff__one__le,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_6282_sum__diff__nat,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6283_sum__diff__nat,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6284_sum__diff__nat,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6285_sum__diff__nat,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_6286_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6287_sum__shift__lb__Suc0__0,axiom,
! [F: nat > rat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_rat )
=> ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6288_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6289_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6290_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6291_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6292_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6293_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6294_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6295_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6296_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6297_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6298_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6299_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6300_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6301_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6302_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6303_remove__induct,axiom,
! [P2: set_Pr4329608150637261639at_nat > $o,B5: set_Pr4329608150637261639at_nat] :
( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ( ~ ( finite4343798906461161616at_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A6 )
=> ( ( A6 != bot_bo228742789529271731at_nat )
=> ( ( ord_le1268244103169919719at_nat @ A6 @ B5 )
=> ( ! [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A6 )
=> ( P2 @ ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6304_remove__induct,axiom,
! [P2: set_set_nat > $o,B5: set_set_nat] :
( ( P2 @ bot_bot_set_set_nat )
=> ( ( ~ ( finite1152437895449049373et_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( A6 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A6 @ B5 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6305_remove__induct,axiom,
! [P2: set_complex > $o,B5: set_complex] :
( ( P2 @ bot_bot_set_complex )
=> ( ( ~ ( finite3207457112153483333omplex @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_complex] :
( ( finite3207457112153483333omplex @ A6 )
=> ( ( A6 != bot_bot_set_complex )
=> ( ( ord_le211207098394363844omplex @ A6 @ B5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A6 )
=> ( P2 @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6306_remove__induct,axiom,
! [P2: set_Pr1261947904930325089at_nat > $o,B5: set_Pr1261947904930325089at_nat] :
( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ( ~ ( finite6177210948735845034at_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A6 )
=> ( ( A6 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A6 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A6 )
=> ( P2 @ ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6307_remove__induct,axiom,
! [P2: set_real > $o,B5: set_real] :
( ( P2 @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_real @ A6 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6308_remove__induct,axiom,
! [P2: set_nat > $o,B5: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6309_remove__induct,axiom,
! [P2: set_int > $o,B5: set_int] :
( ( P2 @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ( A6 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A6 @ B5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_int @ A6 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_6310_finite__remove__induct,axiom,
! [B5: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ B5 )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [A6: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A6 )
=> ( ( A6 != bot_bo228742789529271731at_nat )
=> ( ( ord_le1268244103169919719at_nat @ A6 @ B5 )
=> ( ! [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A6 )
=> ( P2 @ ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6311_finite__remove__induct,axiom,
! [B5: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ B5 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A6: set_set_nat] :
( ( finite1152437895449049373et_nat @ A6 )
=> ( ( A6 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A6 @ B5 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6312_finite__remove__induct,axiom,
! [B5: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [A6: set_complex] :
( ( finite3207457112153483333omplex @ A6 )
=> ( ( A6 != bot_bot_set_complex )
=> ( ( ord_le211207098394363844omplex @ A6 @ B5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A6 )
=> ( P2 @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6313_finite__remove__induct,axiom,
! [B5: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [A6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A6 )
=> ( ( A6 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A6 @ B5 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A6 )
=> ( P2 @ ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6314_finite__remove__induct,axiom,
! [B5: set_real,P2: set_real > $o] :
( ( finite_finite_real @ B5 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_real @ A6 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6315_finite__remove__induct,axiom,
! [B5: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6316_finite__remove__induct,axiom,
! [B5: set_int,P2: set_int > $o] :
( ( finite_finite_int @ B5 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ( A6 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A6 @ B5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( P2 @ ( minus_minus_set_int @ A6 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_6317_finite__induct__select,axiom,
! [S2: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ S2 )
=> ( ( P2 @ bot_bo228742789529271731at_nat )
=> ( ! [T5: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ ( minus_3314409938677909166at_nat @ S2 @ T5 ) )
& ( P2 @ ( insert9069300056098147895at_nat @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6318_finite__induct__select,axiom,
! [S2: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( P2 @ bot_bot_set_complex )
=> ( ! [T5: set_complex] :
( ( ord_less_set_complex @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ S2 @ T5 ) )
& ( P2 @ ( insert_complex @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6319_finite__induct__select,axiom,
! [S2: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ S2 )
=> ( ( P2 @ bot_bo2099793752762293965at_nat )
=> ( ! [T5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ S2 @ T5 ) )
& ( P2 @ ( insert8211810215607154385at_nat @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6320_finite__induct__select,axiom,
! [S2: set_real,P2: set_real > $o] :
( ( finite_finite_real @ S2 )
=> ( ( P2 @ bot_bot_set_real )
=> ( ! [T5: set_real] :
( ( ord_less_set_real @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ S2 @ T5 ) )
& ( P2 @ ( insert_real @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6321_finite__induct__select,axiom,
! [S2: set_int,P2: set_int > $o] :
( ( finite_finite_int @ S2 )
=> ( ( P2 @ bot_bot_set_int )
=> ( ! [T5: set_int] :
( ( ord_less_set_int @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ S2 @ T5 ) )
& ( P2 @ ( insert_int @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6322_finite__induct__select,axiom,
! [S2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S2 )
=> ( ( P2 @ T5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ S2 @ T5 ) )
& ( P2 @ ( insert_nat @ X3 @ T5 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_6323_psubset__insert__iff,axiom,
! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
= ( ( ( member8757157785044589968at_nat @ X @ B5 )
=> ( ord_le2604355607129572851at_nat @ A4 @ B5 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ B5 )
=> ( ( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6324_psubset__insert__iff,axiom,
! [A4: set_set_nat,X: set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
= ( ( ( member_set_nat @ X @ B5 )
=> ( ord_less_set_set_nat @ A4 @ B5 ) )
& ( ~ ( member_set_nat @ X @ B5 )
=> ( ( ( member_set_nat @ X @ A4 )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
& ( ~ ( member_set_nat @ X @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6325_psubset__insert__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
= ( ( ( member8440522571783428010at_nat @ X @ B5 )
=> ( ord_le7866589430770878221at_nat @ A4 @ B5 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6326_psubset__insert__iff,axiom,
! [A4: set_real,X: real,B5: set_real] :
( ( ord_less_set_real @ A4 @ ( insert_real @ X @ B5 ) )
= ( ( ( member_real @ X @ B5 )
=> ( ord_less_set_real @ A4 @ B5 ) )
& ( ~ ( member_real @ X @ B5 )
=> ( ( ( member_real @ X @ A4 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ord_less_eq_set_real @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6327_psubset__insert__iff,axiom,
! [A4: set_nat,X: nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
= ( ( ( member_nat @ X @ B5 )
=> ( ord_less_set_nat @ A4 @ B5 ) )
& ( ~ ( member_nat @ X @ B5 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6328_psubset__insert__iff,axiom,
! [A4: set_int,X: int,B5: set_int] :
( ( ord_less_set_int @ A4 @ ( insert_int @ X @ B5 ) )
= ( ( ( member_int @ X @ B5 )
=> ( ord_less_set_int @ A4 @ B5 ) )
& ( ~ ( member_int @ X @ B5 )
=> ( ( ( member_int @ X @ A4 )
=> ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6329_set__replicate__Suc,axiom,
! [N: nat,X: produc3843707927480180839at_nat] :
( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N ) @ X ) )
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% set_replicate_Suc
thf(fact_6330_set__replicate__Suc,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).
% set_replicate_Suc
thf(fact_6331_set__replicate__Suc,axiom,
! [N: nat,X: product_prod_nat_nat] :
( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N ) @ X ) )
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% set_replicate_Suc
thf(fact_6332_set__replicate__Suc,axiom,
! [N: nat,X: real] :
( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
= ( insert_real @ X @ bot_bot_set_real ) ) ).
% set_replicate_Suc
thf(fact_6333_set__replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_6334_set__replicate__Suc,axiom,
! [N: nat,X: int] :
( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ).
% set_replicate_Suc
thf(fact_6335_set__replicate__conv__if,axiom,
! [N: nat,X: produc3843707927480180839at_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
= bot_bo228742789529271731at_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6336_set__replicate__conv__if,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( ( N = zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= bot_bo8194388402131092736T_VEBT ) )
& ( ( N != zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6337_set__replicate__conv__if,axiom,
! [N: nat,X: product_prod_nat_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6338_set__replicate__conv__if,axiom,
! [N: nat,X: real] :
( ( ( N = zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X ) )
= bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( ( set_real2 @ ( replicate_real @ N @ X ) )
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6339_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6340_set__replicate__conv__if,axiom,
! [N: nat,X: int] :
( ( ( N = zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= bot_bot_set_int ) )
& ( ( N != zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% set_replicate_conv_if
thf(fact_6341_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ M2 )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6342_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ M2 )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6343_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M2 )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6344_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ M2 )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6345_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_6346_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_6347_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_6348_ceiling__split,axiom,
! [P2: int > $o,T: real] :
( ( P2 @ ( archim7802044766580827645g_real @ T ) )
= ( ! [I4: int] :
( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
& ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% ceiling_split
thf(fact_6349_ceiling__split,axiom,
! [P2: int > $o,T: rat] :
( ( P2 @ ( archim2889992004027027881ng_rat @ T ) )
= ( ! [I4: int] :
( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
& ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% ceiling_split
thf(fact_6350_ceiling__eq__iff,axiom,
! [X: real,A: int] :
( ( ( archim7802044766580827645g_real @ X )
= A )
= ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_6351_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim2889992004027027881ng_rat @ X )
= A )
= ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_6352_ceiling__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
=> ( ( archim7802044766580827645g_real @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_6353_ceiling__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_6354_ceiling__correct,axiom,
! [X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% ceiling_correct
thf(fact_6355_ceiling__correct,axiom,
! [X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% ceiling_correct
thf(fact_6356_mult__ceiling__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_6357_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_6358_ceiling__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% ceiling_less_iff
thf(fact_6359_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% ceiling_less_iff
thf(fact_6360_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% le_ceiling_iff
thf(fact_6361_le__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_6362_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > int,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6363_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > rat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6364_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > nat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6365_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > real,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6366_sum_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6367_sum_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6368_sum_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6369_sum_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6370_sum_Oremove,axiom,
! [A4: set_real,X: real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6371_sum_Oremove,axiom,
! [A4: set_real,X: real,G: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups1932886352136224148al_int @ G @ A4 )
= ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6372_sum_Oremove,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups8097168146408367636l_real @ G @ A4 )
= ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6373_sum_Oremove,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6374_sum_Oremove,axiom,
! [A4: set_int,X: int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6375_sum_Oremove,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6376_sum_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > nat,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6377_sum_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > int,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6378_sum_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > real,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6379_sum_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > rat,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A4 ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6380_sum_Oinsert__remove,axiom,
! [A4: set_real,G: real > nat,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6381_sum_Oinsert__remove,axiom,
! [A4: set_real,G: real > int,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6382_sum_Oinsert__remove,axiom,
! [A4: set_real,G: real > real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6383_sum_Oinsert__remove,axiom,
! [A4: set_real,G: real > rat,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6384_sum_Oinsert__remove,axiom,
! [A4: set_int,G: int > nat,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6385_sum_Oinsert__remove,axiom,
! [A4: set_int,G: int > real,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A4 ) )
= ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6386_sum__diff1,axiom,
! [A4: set_complex,A: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups5690904116761175830ex_int @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6387_sum__diff1,axiom,
! [A4: set_real,A: real,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups1932886352136224148al_int @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6388_sum__diff1,axiom,
! [A4: set_complex,A: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6389_sum__diff1,axiom,
! [A4: set_real,A: real,F: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6390_sum__diff1,axiom,
! [A4: set_int,A: int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ A @ A4 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A4 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6391_sum__diff1,axiom,
! [A4: set_complex,A: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6392_sum__diff1,axiom,
! [A4: set_real,A: real,F: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6393_sum__diff1,axiom,
! [A4: set_int,A: int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ A @ A4 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A4 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6394_sum__diff1,axiom,
! [A4: set_nat,A: nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ A @ A4 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3539618377306564664at_int @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6395_sum__diff1,axiom,
! [A4: set_nat,A: nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ A @ A4 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ) ) ).
% sum_diff1
thf(fact_6396_sum__count__set,axiom,
! [Xs: list_complex,X7: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
=> ( ( finite3207457112153483333omplex @ X7 )
=> ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
= ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6397_sum__count__set,axiom,
! [Xs: list_P6011104703257516679at_nat,X7: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X7 )
=> ( ( finite6177210948735845034at_nat @ X7 )
=> ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X7 )
= ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6398_sum__count__set,axiom,
! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
=> ( ( finite5795047828879050333T_VEBT @ X7 )
=> ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6399_sum__count__set,axiom,
! [Xs: list_o,X7: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X7 )
=> ( ( finite_finite_o @ X7 )
=> ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X7 )
= ( size_size_list_o @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6400_sum__count__set,axiom,
! [Xs: list_int,X7: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
=> ( ( finite_finite_int @ X7 )
=> ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
= ( size_size_list_int @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6401_sum__count__set,axiom,
! [Xs: list_nat,X7: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
=> ( ( finite_finite_nat @ X7 )
=> ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_6402_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_6403_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > nat,C2: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6404_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > int,C2: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups5690904116761175830ex_int @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6405_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > real,C2: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups5808333547571424918x_real @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6406_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > rat,C2: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups5058264527183730370ex_rat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6407_sum_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > nat,C2: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups1935376822645274424al_nat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6408_sum_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > int,C2: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1932886352136224148al_int
@ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups1932886352136224148al_int @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6409_sum_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > real,C2: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups8097168146408367636l_real @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6410_sum_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > rat,C2: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups1300246762558778688al_rat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6411_sum_Odelta__remove,axiom,
! [S2: set_int,A: int,B: int > nat,C2: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_nat @ ( B @ A ) @ ( groups4541462559716669496nt_nat @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups4541462559716669496nt_nat @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6412_sum_Odelta__remove,axiom,
! [S2: set_int,A: int,B: int > real,C2: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups8778361861064173332t_real @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_6413_ceiling__divide__upper,axiom,
! [Q2: real,P: real] :
( ( ord_less_real @ zero_zero_real @ Q2 )
=> ( ord_less_eq_real @ P @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P @ Q2 ) ) ) @ Q2 ) ) ) ).
% ceiling_divide_upper
thf(fact_6414_ceiling__divide__upper,axiom,
! [Q2: rat,P: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q2 )
=> ( ord_less_eq_rat @ P @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P @ Q2 ) ) ) @ Q2 ) ) ) ).
% ceiling_divide_upper
thf(fact_6415_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_6416_member__le__sum,axiom,
! [I2: complex,A4: set_complex,F: complex > real] :
( ( member_complex @ I2 @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6417_member__le__sum,axiom,
! [I2: real,A4: set_real,F: real > real] :
( ( member_real @ I2 @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( finite_finite_real @ A4 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6418_member__le__sum,axiom,
! [I2: int,A4: set_int,F: int > real] :
( ( member_int @ I2 @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( finite_finite_int @ A4 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6419_member__le__sum,axiom,
! [I2: complex,A4: set_complex,F: complex > rat] :
( ( member_complex @ I2 @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6420_member__le__sum,axiom,
! [I2: real,A4: set_real,F: real > rat] :
( ( member_real @ I2 @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( finite_finite_real @ A4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6421_member__le__sum,axiom,
! [I2: int,A4: set_int,F: int > rat] :
( ( member_int @ I2 @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( finite_finite_int @ A4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6422_member__le__sum,axiom,
! [I2: nat,A4: set_nat,F: nat > rat] :
( ( member_nat @ I2 @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6423_member__le__sum,axiom,
! [I2: complex,A4: set_complex,F: complex > nat] :
( ( member_complex @ I2 @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6424_member__le__sum,axiom,
! [I2: real,A4: set_real,F: real > nat] :
( ( member_real @ I2 @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( finite_finite_real @ A4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6425_member__le__sum,axiom,
! [I2: int,A4: set_int,F: int > nat] :
( ( member_int @ I2 @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( finite_finite_int @ A4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ) ) ).
% member_le_sum
thf(fact_6426_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > complex] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_complex @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_complex ) ) ) ).
% sum_natinterval_diff
thf(fact_6427_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_6428_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_6429_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_6430_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_6431_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_rat @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_6432_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_6433_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6434_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
= ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6435_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_6436_ceiling__divide__lower,axiom,
! [Q2: rat,P: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q2 )
=> ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P ) ) ).
% ceiling_divide_lower
thf(fact_6437_ceiling__divide__lower,axiom,
! [Q2: real,P: real] :
( ( ord_less_real @ zero_zero_real @ Q2 )
=> ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P ) ) ).
% ceiling_divide_lower
thf(fact_6438_ceiling__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim7802044766580827645g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_6439_ceiling__eq,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_6440_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6441_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6442_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6443_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6444_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6445_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6446_sum_Oin__pairs,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6447_sum_Oin__pairs,axiom,
! [G: nat > rat,M2: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6448_sum_Oin__pairs,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6449_sum_Oin__pairs,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6450_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_6451_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_6452_sum__gp,axiom,
! [N: nat,M2: nat,X: complex] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6453_sum__gp,axiom,
! [N: nat,M2: nat,X: rat] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6454_sum__gp,axiom,
! [N: nat,M2: nat,X: real] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6455_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6456_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6457_sum__gp__offset,axiom,
! [X: complex,M2: nat,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6458_sum__gp__offset,axiom,
! [X: rat,M2: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6459_sum__gp__offset,axiom,
! [X: real,M2: nat,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6460_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6461_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6462_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6463_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6464_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6465_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6466_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6467_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6468_arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_6469_arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_6470_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6471_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6472_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6473_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( numeral_numeral_int @ V ) )
= ( M2
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_6474_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_6475_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6476_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6477_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6478_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_6479_int__dvd__int__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ).
% int_dvd_int_iff
thf(fact_6480_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri8010041392384452111omplex @ M2 )
= zero_zero_complex )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6481_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri681578069525770553at_rat @ M2 )
= zero_zero_rat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6482_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6483_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6484_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6485_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6486_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6487_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6488_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6489_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6490_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_6491_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_6492_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_6493_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_6494_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_6495_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6496_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6497_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6498_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6499_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6500_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6501_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6502_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6503_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_add
thf(fact_6504_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_6505_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_6506_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_6507_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M2 @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_mult
thf(fact_6508_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_6509_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_6510_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_6511_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_6512_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_6513_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_6514_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_6515_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_6516_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_6517_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6518_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6519_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6520_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6521_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6522_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6523_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6524_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6525_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6526_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6527_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_6528_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n3304061248610475627l_real @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6529_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6530_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6531_of__nat__sum,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6532_of__nat__sum,axiom,
! [F: complex > nat,A4: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A4 ) )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( semiri8010041392384452111omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6533_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6534_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( semiri1316708129612266289at_nat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6535_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( semiri5074537144036343181t_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6536_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6537_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6538_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6539_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6540_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6541_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6542_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6543_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6544_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6545_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_6546_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_6547_numeral__le__real__of__nat__iff,axiom,
! [N: num,M2: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_6548_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6549_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6550_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6551_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6552_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6553_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6554_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6555_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6556_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6557_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6558_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6559_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6560_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6561_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6562_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6563_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6564_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6565_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6566_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6567_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6568_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6569_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6570_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6571_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6572_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6573_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6574_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6575_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6576_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6577_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6578_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6579_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6580_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6581_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6582_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6583_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6584_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6585_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6586_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6587_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6588_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_6589_real__arch__simple,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% real_arch_simple
thf(fact_6590_reals__Archimedean2,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% reals_Archimedean2
thf(fact_6591_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_6592_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6593_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6594_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6595_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6596_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6597_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_6598_int__diff__cases,axiom,
! [Z: int] :
~ ! [M: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_6599_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6600_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6601_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6602_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6603_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6604_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6605_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6606_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_6607_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_6608_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_6609_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_6610_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_6611_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_6612_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_6613_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_6614_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_6615_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6616_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6617_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6618_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6619_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6620_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6621_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6622_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6623_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_6624_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_6625_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_6626_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_6627_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_6628_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_6629_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_6630_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_6631_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_6632_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_6633_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_6634_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_6635_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_6636_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_6637_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_6638_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_6639_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_6640_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_6641_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_max
thf(fact_6642_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_6643_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_6644_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_6645_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_6646_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_6647_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6648_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6649_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6650_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6651_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y6: real] :
? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_6652_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_6653_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_6654_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_6655_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_6656_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_6657_int__sum,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_sum
thf(fact_6658_int__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_sum
thf(fact_6659_mod__mult2__eq_H,axiom,
! [A: int,M2: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6660_mod__mult2__eq_H,axiom,
! [A: nat,M2: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6661_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_6662_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_6663_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_6664_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_6665_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_6666_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K2: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_6667_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_6668_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_6669_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_6670_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_6671_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_6672_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_6673_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_6674_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_6675_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6676_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6677_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6678_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6679_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6680_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_6681_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_6682_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_6683_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_6684_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_6685_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_6686_double__gauss__sum,axiom,
! [N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% double_gauss_sum
thf(fact_6687_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_6688_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_6689_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_6690_double__arith__series,axiom,
! [A: complex,D: complex,N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6691_double__arith__series,axiom,
! [A: rat,D: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6692_double__arith__series,axiom,
! [A: extended_enat,D: extended_enat,N: nat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
@ ( groups7108830773950497114d_enat
@ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6693_double__arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6694_double__arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6695_double__arith__series,axiom,
! [A: real,D: real,N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6696_of__nat__code__if,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
@ ( produc1917071388513777916omplex
@ ^ [M3: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6697_of__nat__code__if,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
@ ( produc6207742614233964070at_rat
@ ^ [M3: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6698_of__nat__code__if,axiom,
( semiri4216267220026989637d_enat
= ( ^ [N2: nat] :
( if_Extended_enat @ ( N2 = zero_zero_nat ) @ zero_z5237406670263579293d_enat
@ ( produc2676513652042109336d_enat
@ ^ [M3: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M3 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M3 ) ) @ one_on7984719198319812577d_enat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6699_of__nat__code__if,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
@ ( produc6840382203811409530at_int
@ ^ [M3: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6700_of__nat__code__if,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( produc1703576794950452218t_real
@ ^ [M3: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6701_of__nat__code__if,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( produc6842872674320459806at_nat
@ ^ [M3: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6702_lemma__termdiff3,axiom,
! [H2: real,Z: real,K5: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6703_lemma__termdiff3,axiom,
! [H2: complex,Z: complex,K5: real,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6704_ln__series,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ( ln_ln_real @ X )
= ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_6705_lemma__termdiff2,axiom,
! [H2: complex,Z: complex,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H2
@ ( groups2073611262835488442omplex
@ ^ [P6: nat] :
( groups2073611262835488442omplex
@ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6706_lemma__termdiff2,axiom,
! [H2: rat,Z: rat,N: nat] :
( ( H2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_rat @ H2
@ ( groups2906978787729119204at_rat
@ ^ [P6: nat] :
( groups2906978787729119204at_rat
@ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6707_lemma__termdiff2,axiom,
! [H2: real,Z: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_real @ H2
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] :
( groups6591440286371151544t_real
@ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6708_pochhammer__double,axiom,
! [Z: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6709_pochhammer__double,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6710_pochhammer__double,axiom,
! [Z: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6711_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( semiri2816024913162550771omplex
@ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
@ N2
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_6712_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
@ N2
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_6713_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_6714_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( semiri7260567687927622513x_real
@ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
@ N2
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_6715_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_6716_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_6717_lessThan__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_lessThan_int @ X )
= ( set_ord_lessThan_int @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_6718_lessThan__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( set_or5984915006950818249n_real @ X )
= ( set_or5984915006950818249n_real @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_6719_lessThan__iff,axiom,
! [I2: set_nat,K2: set_nat] :
( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K2 ) )
= ( ord_less_set_nat @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6720_lessThan__iff,axiom,
! [I2: rat,K2: rat] :
( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K2 ) )
= ( ord_less_rat @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6721_lessThan__iff,axiom,
! [I2: num,K2: num] :
( ( member_num @ I2 @ ( set_ord_lessThan_num @ K2 ) )
= ( ord_less_num @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6722_lessThan__iff,axiom,
! [I2: nat,K2: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6723_lessThan__iff,axiom,
! [I2: int,K2: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6724_lessThan__iff,axiom,
! [I2: real,K2: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K2 ) )
= ( ord_less_real @ I2 @ K2 ) ) ).
% lessThan_iff
thf(fact_6725_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).
% finite_lessThan
thf(fact_6726_lessThan__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6727_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6728_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6729_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6730_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6731_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_6732_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_6733_pochhammer__0,axiom,
! [A: real] :
( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_6734_pochhammer__0,axiom,
! [A: complex] :
( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_6735_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_6736_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_6737_single__Diff__lessThan,axiom,
! [K2: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K2 @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K2 ) )
= ( insert_nat @ K2 @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_6738_single__Diff__lessThan,axiom,
! [K2: int] :
( ( minus_minus_set_int @ ( insert_int @ K2 @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K2 ) )
= ( insert_int @ K2 @ bot_bot_set_int ) ) ).
% single_Diff_lessThan
thf(fact_6739_single__Diff__lessThan,axiom,
! [K2: real] :
( ( minus_minus_set_real @ ( insert_real @ K2 @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K2 ) )
= ( insert_real @ K2 @ bot_bot_set_real ) ) ).
% single_Diff_lessThan
thf(fact_6740_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6741_sum_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6742_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6743_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6744_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_6745_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_6746_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_6747_lessThan__non__empty,axiom,
! [X: int] :
( ( set_ord_lessThan_int @ X )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_6748_lessThan__non__empty,axiom,
! [X: real] :
( ( set_or5984915006950818249n_real @ X )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_6749_infinite__Iio,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% infinite_Iio
thf(fact_6750_infinite__Iio,axiom,
! [A: real] :
~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% infinite_Iio
thf(fact_6751_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] : ( ord_less_set_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6752_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X5: rat] : ( ord_less_rat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6753_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X5: num] : ( ord_less_num @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6754_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6755_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_int @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6756_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_real @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6757_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_6758_lessThan__strict__subset__iff,axiom,
! [M2: rat,N: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M2 ) @ ( set_ord_lessThan_rat @ N ) )
= ( ord_less_rat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6759_lessThan__strict__subset__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6760_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6761_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6762_lessThan__strict__subset__iff,axiom,
! [M2: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6763_pochhammer__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_6764_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_6765_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_6766_pochhammer__pos,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_6767_pochhammer__eq__0__mono,axiom,
! [A: complex,N: nat,M2: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s2602460028002588243omplex @ A @ M2 )
= zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_6768_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M2: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s7457072308508201937r_real @ A @ M2 )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_6769_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M2: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s4028243227959126397er_rat @ A @ M2 )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_6770_pochhammer__neq__0__mono,axiom,
! [A: complex,M2: nat,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ M2 )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s2602460028002588243omplex @ A @ N )
!= zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_6771_pochhammer__neq__0__mono,axiom,
! [A: real,M2: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M2 )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s7457072308508201937r_real @ A @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_6772_pochhammer__neq__0__mono,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M2 )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s4028243227959126397er_rat @ A @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_6773_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_6774_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_6775_finite__nat__bounded,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [K: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K ) ) ) ).
% finite_nat_bounded
thf(fact_6776_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S7: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_6777_pochhammer__nonneg,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_6778_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_6779_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_6780_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_6781_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_6782_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_6783_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_6784_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_6785_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_6786_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_6787_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_6788_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_6789_sum__diff__distrib,axiom,
! [Q: int > nat,P2: int > nat,N: int] :
( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P2 @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_ord_lessThan_int @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6790_sum__diff__distrib,axiom,
! [Q: real > nat,P2: real > nat,N: real] :
( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P2 @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X5: real] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6791_sum__diff__distrib,axiom,
! [Q: nat > nat,P2: nat > nat,N: nat] :
( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6792_pochhammer__rec,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_6793_pochhammer__rec,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_6794_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_6795_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_6796_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_6797_pochhammer__Suc,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_6798_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_6799_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_6800_pochhammer__Suc,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_6801_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_6802_pochhammer__rec_H,axiom,
! [Z: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
= ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_6803_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_6804_pochhammer__rec_H,axiom,
! [Z: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_6805_pochhammer__rec_H,axiom,
! [Z: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_6806_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_6807_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_6808_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_6809_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_6810_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_6811_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
= zero_zero_complex )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_6812_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
= zero_zero_rat )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_6813_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
= zero_zero_int )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_6814_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
= zero_zero_real )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_6815_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_6816_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_6817_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_6818_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_6819_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_6820_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_6821_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_6822_pochhammer__product_H,axiom,
! [Z: complex,N: nat,M2: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_6823_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M2: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_6824_pochhammer__product_H,axiom,
! [Z: int,N: nat,M2: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_6825_pochhammer__product_H,axiom,
! [Z: real,N: nat,M2: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_6826_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M2: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_6827_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6828_sum_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6829_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6830_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6831_sum__lessThan__telescope,axiom,
! [F: nat > int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6832_sum__lessThan__telescope,axiom,
! [F: nat > rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_rat @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6833_sum__lessThan__telescope,axiom,
! [F: nat > real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6834_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6835_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6836_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6837_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_6838_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_6839_one__diff__power__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6840_one__diff__power__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6841_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6842_one__diff__power__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6843_power__diff__1__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
= ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6844_power__diff__1__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6845_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6846_power__diff__1__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6847_geometric__sum,axiom,
! [X: complex,N: nat] :
( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% geometric_sum
thf(fact_6848_geometric__sum,axiom,
! [X: rat,N: nat] :
( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_6849_geometric__sum,axiom,
! [X: real,N: nat] :
( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% geometric_sum
thf(fact_6850_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s2602460028002588243omplex @ Z @ N )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_6851_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_6852_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4660882817536571857er_int @ Z @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_6853_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s7457072308508201937r_real @ Z @ N )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_6854_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_6855_sum__gp__strict,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6856_sum__gp__strict,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6857_sum__gp__strict,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6858_lemma__termdiff1,axiom,
! [Z: complex,H2: complex,M2: nat] :
( ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z @ P6 ) ) @ ( power_power_complex @ Z @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6859_lemma__termdiff1,axiom,
! [Z: rat,H2: rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_rat @ Z @ P6 ) ) @ ( power_power_rat @ Z @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P6 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6860_lemma__termdiff1,axiom,
! [Z: int,H2: int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z @ P6 ) ) @ ( power_power_int @ Z @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6861_lemma__termdiff1,axiom,
! [Z: real,H2: real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z @ P6 ) ) @ ( power_power_real @ Z @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6862_power__diff__sumr2,axiom,
! [X: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6863_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6864_power__diff__sumr2,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6865_power__diff__sumr2,axiom,
! [X: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6866_diff__power__eq__sum,axiom,
! [X: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X @ P6 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6867_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X @ P6 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6868_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6869_diff__power__eq__sum,axiom,
! [X: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X @ P6 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6870_pochhammer__absorb__comp,axiom,
! [R2: complex,K2: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K2 ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_6871_pochhammer__absorb__comp,axiom,
! [R2: rat,K2: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K2 ) )
= ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_6872_pochhammer__absorb__comp,axiom,
! [R2: int,K2: nat] :
( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K2 ) )
= ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_6873_pochhammer__absorb__comp,axiom,
! [R2: real,K2: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K2 ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_6874_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > rat,K5: rat,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6875_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K5: int,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K5 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6876_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K5: nat,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6877_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K5: real,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K5 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6878_one__diff__power__eq_H,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6879_one__diff__power__eq_H,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6880_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6881_one__diff__power__eq_H,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6882_pochhammer__minus_H,axiom,
! [B: complex,K2: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_6883_pochhammer__minus_H,axiom,
! [B: rat,K2: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_6884_pochhammer__minus_H,axiom,
! [B: int,K2: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_6885_pochhammer__minus_H,axiom,
! [B: real,K2: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_6886_pochhammer__minus,axiom,
! [B: complex,K2: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_6887_pochhammer__minus,axiom,
! [B: rat,K2: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_6888_pochhammer__minus,axiom,
! [B: int,K2: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_6889_pochhammer__minus,axiom,
! [B: real,K2: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_6890_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_6891_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_6892_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_6893_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_6894_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_6895_suminf__geometric,axiom,
! [C2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C2 ) @ one_one_real )
=> ( ( suminf_real @ ( power_power_real @ C2 ) )
= ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C2 ) ) ) ) ).
% suminf_geometric
thf(fact_6896_suminf__geometric,axiom,
! [C2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C2 ) @ one_one_real )
=> ( ( suminf_complex @ ( power_power_complex @ C2 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C2 ) ) ) ) ).
% suminf_geometric
thf(fact_6897_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_6898_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_6899_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_6900_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_6901_suminf__zero,axiom,
( ( suminf_complex
@ ^ [N2: nat] : zero_zero_complex )
= zero_zero_complex ) ).
% suminf_zero
thf(fact_6902_suminf__zero,axiom,
( ( suminf_real
@ ^ [N2: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_6903_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N2: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_6904_suminf__zero,axiom,
( ( suminf_int
@ ^ [N2: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_6905_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_6906_norm__uminus__minus,axiom,
! [X: real,Y: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
= ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_6907_norm__uminus__minus,axiom,
! [X: complex,Y: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
= ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_6908_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_6909_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_6910_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_6911_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_6912_norm__mult__less,axiom,
! [X: real,R2: real,Y: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% norm_mult_less
thf(fact_6913_norm__mult__less,axiom,
! [X: complex,R2: real,Y: complex,S: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% norm_mult_less
thf(fact_6914_norm__triangle__lt,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_6915_norm__triangle__lt,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_6916_norm__add__less,axiom,
! [X: real,R2: real,Y: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_add_less
thf(fact_6917_norm__add__less,axiom,
! [X: complex,R2: real,Y: complex,S: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_add_less
thf(fact_6918_norm__add__leD,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C2 ) ) ) ).
% norm_add_leD
thf(fact_6919_norm__add__leD,axiom,
! [A: complex,B: complex,C2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C2 ) ) ) ).
% norm_add_leD
thf(fact_6920_norm__triangle__le,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_6921_norm__triangle__le,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_6922_norm__triangle__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_6923_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_6924_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_6925_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_6926_norm__diff__triangle__less,axiom,
! [X: real,Y: real,E1: real,Z: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_6927_norm__diff__triangle__less,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_6928_norm__diff__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_6929_norm__diff__ineq,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_6930_suminf__finite,axiom,
! [N6: set_nat,F: nat > complex] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_complex ) )
=> ( ( suminf_complex @ F )
= ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_6931_suminf__finite,axiom,
! [N6: set_nat,F: nat > int] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_int ) )
=> ( ( suminf_int @ F )
= ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_6932_suminf__finite,axiom,
! [N6: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_nat ) )
=> ( ( suminf_nat @ F )
= ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_6933_suminf__finite,axiom,
! [N6: set_nat,F: nat > real] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_real ) )
=> ( ( suminf_real @ F )
= ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% suminf_finite
thf(fact_6934_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_6935_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_6936_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C2: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C2 @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_6937_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C2: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C2 @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_6938_pochhammer__times__pochhammer__half,axiom,
! [Z: complex,N: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_6939_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_6940_pochhammer__times__pochhammer__half,axiom,
! [Z: real,N: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups129246275422532515t_real
@ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_6941_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_6942_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L2: int] :
( if_int
@ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_6943_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_6944_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A5: complex,N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_6945_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A5: rat,N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_6946_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A5: int,N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_6947_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_6948_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_6949_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_6950_and__zero__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% and_zero_eq
thf(fact_6951_and__zero__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% and_zero_eq
thf(fact_6952_zero__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% zero_and_eq
thf(fact_6953_zero__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_and_eq
thf(fact_6954_bit_Oconj__zero__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% bit.conj_zero_left
thf(fact_6955_bit_Oconj__zero__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
= zero_zero_int ) ).
% bit.conj_zero_right
thf(fact_6956_of__nat__prod,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_6957_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( semiri5074537144036343181t_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_6958_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_6959_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups708209901874060359at_nat
@ ^ [X5: nat] : ( semiri1316708129612266289at_nat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_6960_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6961_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups73079841787564623at_rat
@ ^ [X5: nat] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6962_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6963_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups2316167850115554303t_real
@ ^ [X5: int] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6964_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X5: int] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6965_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_6966_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups6464643781859351333omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6967_prod__zero__iff,axiom,
! [A4: set_int,F: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups7440179247065528705omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6968_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups3708469109370488835omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6969_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6970_prod__zero__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6971_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6972_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6973_prod__zero__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6974_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6975_prod__zero__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_6976_prod_Oempty,axiom,
! [G: real > nat] :
( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
= one_one_nat ) ).
% prod.empty
thf(fact_6977_prod_Oempty,axiom,
! [G: real > int] :
( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
= one_one_int ) ).
% prod.empty
thf(fact_6978_prod_Oempty,axiom,
! [G: real > real] :
( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
= one_one_real ) ).
% prod.empty
thf(fact_6979_prod_Oempty,axiom,
! [G: real > complex] :
( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
= one_one_complex ) ).
% prod.empty
thf(fact_6980_prod_Oempty,axiom,
! [G: real > rat] :
( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
= one_one_rat ) ).
% prod.empty
thf(fact_6981_prod_Oempty,axiom,
! [G: nat > real] :
( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
= one_one_real ) ).
% prod.empty
thf(fact_6982_prod_Oempty,axiom,
! [G: nat > complex] :
( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
= one_one_complex ) ).
% prod.empty
thf(fact_6983_prod_Oempty,axiom,
! [G: nat > rat] :
( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
= one_one_rat ) ).
% prod.empty
thf(fact_6984_prod_Oempty,axiom,
! [G: int > nat] :
( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
= one_one_nat ) ).
% prod.empty
thf(fact_6985_prod_Oempty,axiom,
! [G: int > real] :
( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
= one_one_real ) ).
% prod.empty
thf(fact_6986_prod_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G @ A4 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_6987_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_6988_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= one_one_int ) ) ).
% prod.infinite
thf(fact_6989_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_6990_prod_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_6991_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_6992_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_6993_prod_Oinfinite,axiom,
! [A4: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_6994_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_6995_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_6996_dvd__prod__eqI,axiom,
! [A4: set_real,A: real,B: nat,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_6997_dvd__prod__eqI,axiom,
! [A4: set_int,A: int,B: nat,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_6998_dvd__prod__eqI,axiom,
! [A4: set_complex,A: complex,B: nat,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_6999_dvd__prod__eqI,axiom,
! [A4: set_real,A: real,B: int,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7000_dvd__prod__eqI,axiom,
! [A4: set_complex,A: complex,B: int,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7001_dvd__prod__eqI,axiom,
! [A4: set_nat,A: nat,B: int,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7002_dvd__prod__eqI,axiom,
! [A4: set_int,A: int,B: int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7003_dvd__prod__eqI,axiom,
! [A4: set_nat,A: nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7004_dvd__prod__eqI,axiom,
! [A4: set_set_nat,A: set_nat,B: nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups4248547760180025341at_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7005_dvd__prod__eqI,axiom,
! [A4: set_set_nat,A: set_nat,B: int,F: set_nat > int] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups4246057289670975065at_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7006_dvd__prodI,axiom,
! [A4: set_real,A: real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7007_dvd__prodI,axiom,
! [A4: set_int,A: int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7008_dvd__prodI,axiom,
! [A4: set_complex,A: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7009_dvd__prodI,axiom,
! [A4: set_real,A: real,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7010_dvd__prodI,axiom,
! [A4: set_complex,A: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7011_dvd__prodI,axiom,
! [A4: set_nat,A: nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7012_dvd__prodI,axiom,
! [A4: set_int,A: int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7013_dvd__prodI,axiom,
! [A4: set_nat,A: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7014_dvd__prodI,axiom,
! [A4: set_set_nat,A: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups4248547760180025341at_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7015_dvd__prodI,axiom,
! [A4: set_set_nat,A: set_nat,F: set_nat > int] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups4246057289670975065at_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7016_and__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% and_negative_int_iff
thf(fact_7017_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_7018_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K3: int] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K3: int] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_7019_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups861055069439313189ex_nat
@ ^ [K3: complex] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups861055069439313189ex_nat
@ ^ [K3: complex] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_7020_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4694064378042380927al_int
@ ^ [K3: real] : ( if_int @ ( A = K3 ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4694064378042380927al_int
@ ^ [K3: real] : ( if_int @ ( A = K3 ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= one_one_int ) ) ) ) ).
% prod.delta'
thf(fact_7021_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups858564598930262913ex_int
@ ^ [K3: complex] : ( if_int @ ( A = K3 ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups858564598930262913ex_int
@ ^ [K3: complex] : ( if_int @ ( A = K3 ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= one_one_int ) ) ) ) ).
% prod.delta'
thf(fact_7022_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7023_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7024_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7025_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7026_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7027_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4696554848551431203al_nat
@ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_7028_prod_Odelta,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_7029_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups861055069439313189ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups861055069439313189ex_nat
@ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_nat )
@ S2 )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_7030_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4694064378042380927al_int
@ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4694064378042380927al_int
@ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= one_one_int ) ) ) ) ).
% prod.delta
thf(fact_7031_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups858564598930262913ex_int
@ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups858564598930262913ex_int
@ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ one_one_int )
@ S2 )
= one_one_int ) ) ) ) ).
% prod.delta
thf(fact_7032_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7033_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7034_prod_Odelta,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7035_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7036_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7037_prod_Oinsert,axiom,
! [A4: set_real,X: real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7038_prod_Oinsert,axiom,
! [A4: set_nat,X: nat,G: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7039_prod_Oinsert,axiom,
! [A4: set_int,X: int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ~ ( member_int @ X @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7040_prod_Oinsert,axiom,
! [A4: set_complex,X: complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ~ ( member_complex @ X @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7041_prod_Oinsert,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7042_prod_Oinsert,axiom,
! [A4: set_nat,X: nat,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X @ A4 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7043_prod_Oinsert,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ~ ( member_int @ X @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7044_prod_Oinsert,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ~ ( member_complex @ X @ A4 )
=> ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7045_prod_Oinsert,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ~ ( member_real @ X @ A4 )
=> ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7046_prod_Oinsert,axiom,
! [A4: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% prod.insert
thf(fact_7047_prod_OlessThan__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7048_prod_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7049_prod_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7050_prod_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7051_prod_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7052_zero__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_7053_log__less__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_7054_one__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_real @ A @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_7055_log__less__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_real @ X @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_7056_log__less__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_7057_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_7058_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= zero_zero_int ) ).
% and_numerals(5)
thf(fact_7059_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_7060_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= zero_zero_int ) ).
% and_numerals(1)
thf(fact_7061_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_7062_log__le__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_7063_log__le__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_7064_one__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ A @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_7065_log__le__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_7066_zero__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_7067_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7068_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7069_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7070_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7071_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7072_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_7073_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_7074_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_7075_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > int,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X5: real] :
( groups705719431365010083at_int @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y5: nat] :
( groups4694064378042380927al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7076_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > int,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X5: complex] :
( groups705719431365010083at_int @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y5: nat] :
( groups858564598930262913ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7077_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X5: real] :
( groups1705073143266064639nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y5: int] :
( groups4694064378042380927al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7078_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X5: complex] :
( groups1705073143266064639nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y5: int] :
( groups858564598930262913ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7079_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4696554848551431203al_nat
@ ^ [X5: real] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups4696554848551431203al_nat
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7080_prod_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [X5: int] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups1707563613775114915nt_nat
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7081_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups861055069439313189ex_nat
@ ^ [X5: complex] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups861055069439313189ex_nat
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7082_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_real,G: nat > real > int,R: nat > real > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_real @ B5 )
=> ( ( groups705719431365010083at_int
@ ^ [X5: nat] :
( groups4694064378042380927al_int @ ( G @ X5 )
@ ( collect_real
@ ^ [Y5: real] :
( ( member_real @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4694064378042380927al_int
@ ^ [Y5: real] :
( groups705719431365010083at_int
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7083_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > int,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups705719431365010083at_int
@ ^ [X5: nat] :
( groups858564598930262913ex_int @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups858564598930262913ex_int
@ ^ [Y5: complex] :
( groups705719431365010083at_int
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7084_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_nat,G: nat > nat > int,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups705719431365010083at_int
@ ^ [X5: nat] :
( groups705719431365010083at_int @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y5: nat] :
( groups705719431365010083at_int
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7085_prod__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7086_prod__mono,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7087_prod__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ! [I3: int] :
( ( member_int @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7088_prod__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7089_prod__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7090_prod__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ! [I3: int] :
( ( member_int @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7091_prod__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7092_prod__mono,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7093_prod__mono,axiom,
! [A4: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
& ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7094_prod__mono,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
& ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7095_prod__nonneg,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7096_prod__nonneg,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7097_prod__nonneg,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7098_prod__pos,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7099_prod__pos,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7100_prod__pos,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7101_prod__ge__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7102_prod__ge__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7103_prod__ge__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7104_prod__ge__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7105_prod__ge__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7106_prod__ge__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7107_prod__ge__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7108_prod__ge__1,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7109_prod__ge__1,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7110_prod__ge__1,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7111_prod__zero,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_complex ) )
=> ( ( groups6464643781859351333omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7112_prod__zero,axiom,
! [A4: set_int,F: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_complex ) )
=> ( ( groups7440179247065528705omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7113_prod__zero,axiom,
! [A4: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_complex ) )
=> ( ( groups3708469109370488835omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7114_prod__zero,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7115_prod__zero,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7116_prod__zero,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7117_prod__zero,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) )
=> ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7118_prod__zero,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) )
=> ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7119_prod__zero,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X3: complex] :
( ( member_complex @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_rat ) )
=> ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7120_prod__zero,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( ( F @ X3 )
= zero_zero_nat ) )
=> ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_7121_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
@ A
@ B
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_7122_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
@ A
@ B
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_7123_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7124_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_7125_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7126_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
% pi_gt_zero
thf(fact_7127_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
% pi_not_less_zero
thf(fact_7128_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > nat,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups4696554848551431203al_nat @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups4696554848551431203al_nat
@ ^ [X5: real] : ( if_nat @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_nat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7129_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > nat,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups1707563613775114915nt_nat
@ ^ [X5: int] : ( if_nat @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_nat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7130_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > nat,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups861055069439313189ex_nat
@ ^ [X5: complex] : ( if_nat @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_nat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7131_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > int,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups4694064378042380927al_int @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups4694064378042380927al_int
@ ^ [X5: real] : ( if_int @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_int )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7132_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > int,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups858564598930262913ex_int
@ ^ [X5: complex] : ( if_int @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_int )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7133_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups1681761925125756287l_real
@ ^ [X5: real] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7134_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > real,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7135_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2316167850115554303t_real
@ ^ [X5: int] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7136_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups766887009212190081x_real
@ ^ [X5: complex] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7137_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > complex,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups713298508707869441omplex
@ ^ [X5: real] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7138_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7139_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7140_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M2: nat,K2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7141_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7142_prod__le__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7143_prod__le__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7144_prod__le__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7145_prod__le__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7146_prod__le__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7147_prod__le__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7148_prod__le__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7149_prod__le__1,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7150_prod__le__1,axiom,
! [A4: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
& ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_7151_prod__le__1,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
& ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A4 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_7152_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7153_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7154_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7155_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7156_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7157_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7158_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7159_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7160_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7161_prod_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7162_prod_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A4 ) )
= ( groups713298508707869441omplex @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7163_prod_Oinsert__if,axiom,
! [A4: set_nat,X: nat,G: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A4 ) )
= ( groups6464643781859351333omplex @ G @ A4 ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7164_prod_Oinsert__if,axiom,
! [A4: set_int,X: int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ X @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A4 ) )
= ( groups7440179247065528705omplex @ G @ A4 ) ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7165_prod_Oinsert__if,axiom,
! [A4: set_complex,X: complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ X @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A4 ) )
= ( groups3708469109370488835omplex @ G @ A4 ) ) )
& ( ~ ( member_complex @ X @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7166_prod_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A4 ) )
= ( groups1681761925125756287l_real @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7167_prod_Oinsert__if,axiom,
! [A4: set_nat,X: nat,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A4 ) )
= ( groups129246275422532515t_real @ G @ A4 ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7168_prod_Oinsert__if,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( member_int @ X @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A4 ) )
= ( groups2316167850115554303t_real @ G @ A4 ) ) )
& ( ~ ( member_int @ X @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7169_prod_Oinsert__if,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( member_complex @ X @ A4 )
=> ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( groups766887009212190081x_real @ G @ A4 ) ) )
& ( ~ ( member_complex @ X @ A4 )
=> ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7170_prod_Oinsert__if,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( ( member_real @ X @ A4 )
=> ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( groups4061424788464935467al_rat @ G @ A4 ) ) )
& ( ~ ( member_real @ X @ A4 )
=> ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7171_prod_Oinsert__if,axiom,
! [A4: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A4 ) )
= ( groups73079841787564623at_rat @ G @ A4 ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7172_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7173_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7174_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7175_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7176_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7177_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7178_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7179_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7180_prod__dvd__prod__subset2,axiom,
! [B5: set_set_nat,A4: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ( finite1152437895449049373et_nat @ B5 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups4248547760180025341at_nat @ F @ A4 ) @ ( groups4248547760180025341at_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7181_prod__dvd__prod__subset2,axiom,
! [B5: set_set_nat,A4: set_set_nat,F: set_nat > int,G: set_nat > int] :
( ( finite1152437895449049373et_nat @ B5 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups4246057289670975065at_int @ F @ A4 ) @ ( groups4246057289670975065at_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7182_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7183_prod__dvd__prod__subset,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7184_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7185_prod__dvd__prod__subset,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups4075276357253098568at_int @ F @ A4 ) @ ( groups4075276357253098568at_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7186_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7187_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7188_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7189_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7190_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I2: real > real,J: real > real,T3: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7191_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I2: int > real,J: real > int,T3: set_int,G: real > nat,H2: int > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7192_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_complex,S2: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > nat,H2: complex > nat] :
( ( finite_finite_real @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7193_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I2: real > int,J: int > real,T3: set_real,G: int > nat,H2: real > nat] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7194_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I2: int > int,J: int > int,T3: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S2 )
= ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7195_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_complex,S2: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > nat,H2: complex > nat] :
( ( finite_finite_int @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7196_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_real,S2: set_complex,I2: real > complex,J: complex > real,T3: set_real,G: complex > nat,H2: real > nat] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7197_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_int,S2: set_complex,I2: int > complex,J: complex > int,T3: set_int,G: complex > nat,H2: int > nat] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
=> ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7198_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_complex,S2: set_complex,I2: complex > complex,J: complex > complex,T3: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
=> ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7199_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I2: real > real,J: real > real,T3: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
=> ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ S2 )
= ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7200_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_7201_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_7202_and__less__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ K2 ) ) ).
% and_less_eq
thf(fact_7203_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups4696554848551431203al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_nat ) ) ) )
= ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7204_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_nat ) ) ) )
= ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7205_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_nat ) ) ) )
= ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7206_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( groups4694064378042380927al_int @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_int ) ) ) )
= ( groups4694064378042380927al_int @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7207_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_int ) ) ) )
= ( groups858564598930262913ex_int @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7208_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7209_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7210_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7211_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_complex ) ) ) )
= ( groups713298508707869441omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7212_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_complex ) ) ) )
= ( groups7440179247065528705omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7213_prod_Onat__diff__reindex,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_7214_prod_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_7215_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N: nat,M2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7216_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7217_less__1__prod2,axiom,
! [I6: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7218_less__1__prod2,axiom,
! [I6: set_nat,I2: nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I2 @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7219_less__1__prod2,axiom,
! [I6: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I2 @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7220_less__1__prod2,axiom,
! [I6: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7221_less__1__prod2,axiom,
! [I6: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7222_less__1__prod2,axiom,
! [I6: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I2 @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7223_less__1__prod2,axiom,
! [I6: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I2 @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7224_less__1__prod2,axiom,
! [I6: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7225_less__1__prod2,axiom,
! [I6: set_real,I2: real,F: real > int] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I2 @ I6 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7226_less__1__prod2,axiom,
! [I6: set_complex,I2: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I2 @ I6 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7227_less__1__prod,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7228_less__1__prod,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7229_less__1__prod,axiom,
! [I6: set_nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7230_less__1__prod,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7231_less__1__prod,axiom,
! [I6: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7232_less__1__prod,axiom,
! [I6: set_real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7233_less__1__prod,axiom,
! [I6: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7234_less__1__prod,axiom,
! [I6: set_int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7235_less__1__prod,axiom,
! [I6: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7236_less__1__prod,axiom,
! [I6: set_real,F: real > int] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7237_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > complex] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= ( times_times_complex @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups3708469109370488835omplex @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7238_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > complex] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ A4 )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups6464643781859351333omplex @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7239_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7240_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7241_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7242_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7243_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7244_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7245_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > complex] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ A4 )
= ( times_times_complex @ ( groups7440179247065528705omplex @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups7440179247065528705omplex @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7246_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7247_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H2 @ B5 ) )
= ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7248_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H2 @ B5 ) )
= ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7249_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H2 @ B5 ) )
= ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7250_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H2 @ B5 ) )
= ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7251_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) )
= ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7252_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H2 @ B5 ) )
= ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7253_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H2 @ B5 ) )
= ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7254_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) )
= ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7255_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) )
= ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7256_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) )
= ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7257_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) )
=> ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7258_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7259_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) )
=> ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7260_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7261_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) )
=> ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7262_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7263_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) )
=> ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7264_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7265_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) )
=> ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7266_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7267_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7268_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7269_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7270_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7271_prod_Omono__neutral__left,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7272_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ S2 )
= ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7273_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ S2 )
= ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7274_prod_Omono__neutral__left,axiom,
! [T3: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ S2 )
= ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7275_prod_Omono__neutral__left,axiom,
! [T3: set_int,S2: set_int,G: int > nat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S2 )
= ( groups1707563613775114915nt_nat @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7276_prod_Omono__neutral__left,axiom,
! [T3: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ S2 )
= ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7277_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ T3 )
= ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7278_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ T3 )
= ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7279_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ T3 )
= ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7280_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ T3 )
= ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7281_prod_Omono__neutral__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ T3 )
= ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7282_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ T3 )
= ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7283_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ T3 )
= ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7284_prod_Omono__neutral__right,axiom,
! [T3: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T3 )
=> ( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ T3 )
= ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7285_prod_Omono__neutral__right,axiom,
! [T3: set_int,S2: set_int,G: int > nat] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups1707563613775114915nt_nat @ G @ T3 )
= ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7286_prod_Omono__neutral__right,axiom,
! [T3: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T3 )
=> ( ( ord_less_eq_set_int @ S2 @ T3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ T3 )
= ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7287_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7288_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7289_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ S2 )
= ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7290_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7291_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S2 )
= ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7292_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7293_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7294_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7295_prod_Omono__neutral__cong__left,axiom,
! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ S2 )
= ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7296_prod_Omono__neutral__cong__left,axiom,
! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7297_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ T3 )
= ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7298_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ T3 )
= ( groups861055069439313189ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7299_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ T3 )
= ( groups4694064378042380927al_int @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7300_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ T3 )
= ( groups858564598930262913ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7301_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ T3 )
= ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7302_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ T3 )
= ( groups766887009212190081x_real @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7303_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ T3 )
= ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7304_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ T3 )
= ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7305_prod_Omono__neutral__cong__right,axiom,
! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T3 )
=> ( ( ord_less_eq_set_real @ S2 @ T3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ T3 )
= ( groups4061424788464935467al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7306_prod_Omono__neutral__cong__right,axiom,
! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T3 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T3 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ T3 )
= ( groups225925009352817453ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7307_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7308_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7309_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7310_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7311_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7312_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_complex @ ( G @ M2 ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7313_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_real @ ( G @ M2 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7314_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7315_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7316_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7317_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_complex @ ( G @ ( suc @ N ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7318_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7319_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7320_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7321_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7322_log__base__change,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ B @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% log_base_change
thf(fact_7323_log__of__power__eq,axiom,
! [M2: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_7324_less__log__of__power,axiom,
! [B: real,N: nat,M2: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_7325_pi__less__4,axiom,
ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% pi_less_4
thf(fact_7326_prod_OlessThan__Suc__shift,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_complex @ ( G @ zero_zero_nat )
@ ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7327_prod_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7328_prod_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7329_prod_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7330_prod_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7331_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_complex @ ( G @ M2 )
@ ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7332_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ M2 )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7333_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ M2 )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7334_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ M2 )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7335_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ M2 )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7336_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_7337_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_7338_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7339_prod__mono__strict,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7340_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7341_prod__mono__strict,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7342_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7343_prod__mono__strict,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7344_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7345_prod__mono__strict,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7346_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7347_prod__mono__strict,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7348_even__prod__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7349_even__prod__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7350_even__prod__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups4077766827762148844at_nat @ F @ A4 ) )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7351_even__prod__iff,axiom,
! [A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7352_even__prod__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups4075276357253098568at_int @ F @ A4 ) )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7353_even__prod__iff,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7354_even__prod__iff,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7355_even__prod__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7356_prod_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > complex,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7357_prod_Oinsert__remove,axiom,
! [A4: set_real,G: real > complex,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7358_prod_Oinsert__remove,axiom,
! [A4: set_int,G: int > complex,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7359_prod_Oinsert__remove,axiom,
! [A4: set_nat,G: nat > complex,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7360_prod_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > real,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7361_prod_Oinsert__remove,axiom,
! [A4: set_real,G: real > real,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7362_prod_Oinsert__remove,axiom,
! [A4: set_int,G: int > real,X: int] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7363_prod_Oinsert__remove,axiom,
! [A4: set_nat,G: nat > real,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A4 ) )
= ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7364_prod_Oinsert__remove,axiom,
! [A4: set_complex,G: complex > rat,X: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7365_prod_Oinsert__remove,axiom,
! [A4: set_real,G: real > rat,X: real] :
( ( finite_finite_real @ A4 )
=> ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A4 ) )
= ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7366_prod_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7367_prod_Oremove,axiom,
! [A4: set_real,X: real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups713298508707869441omplex @ G @ A4 )
= ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7368_prod_Oremove,axiom,
! [A4: set_int,X: int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ A4 )
= ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7369_prod_Oremove,axiom,
! [A4: set_nat,X: nat,G: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ A4 )
= ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7370_prod_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7371_prod_Oremove,axiom,
! [A4: set_real,X: real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups1681761925125756287l_real @ G @ A4 )
= ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7372_prod_Oremove,axiom,
! [A4: set_int,X: int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ X @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7373_prod_Oremove,axiom,
! [A4: set_nat,X: nat,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7374_prod_Oremove,axiom,
! [A4: set_complex,X: complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ X @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( times_times_rat @ ( G @ X ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7375_prod_Oremove,axiom,
! [A4: set_real,X: real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ X @ A4 )
=> ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7376_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > complex,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7377_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > real,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7378_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > rat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7379_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > int,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7380_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > nat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7381_log__mult,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_7382_log__divide,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_7383_le__log__of__power,axiom,
! [B: real,N: nat,M2: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_7384_log__base__pow,axiom,
! [A: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ ( power_power_real @ A @ N ) @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_7385_log__nat__power,axiom,
! [X: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% log_nat_power
thf(fact_7386_prod_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > complex,C2: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_complex @ ( B @ A ) @ ( groups3708469109370488835omplex @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups3708469109370488835omplex @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7387_prod_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > complex,C2: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_complex @ ( B @ A ) @ ( groups713298508707869441omplex @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups713298508707869441omplex @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7388_prod_Odelta__remove,axiom,
! [S2: set_int,A: int,B: int > complex,C2: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_complex @ ( B @ A ) @ ( groups7440179247065528705omplex @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups7440179247065528705omplex @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7389_prod_Odelta__remove,axiom,
! [S2: set_nat,A: nat,B: nat > complex,C2: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_complex @ ( B @ A ) @ ( groups6464643781859351333omplex @ C2 @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups6464643781859351333omplex @ C2 @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7390_prod_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > real,C2: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups766887009212190081x_real @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7391_prod_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > real,C2: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_real @ ( B @ A ) @ ( groups1681761925125756287l_real @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups1681761925125756287l_real @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7392_prod_Odelta__remove,axiom,
! [S2: set_int,A: int,B: int > real,C2: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_real @ ( B @ A ) @ ( groups2316167850115554303t_real @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups2316167850115554303t_real @ C2 @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7393_prod_Odelta__remove,axiom,
! [S2: set_nat,A: nat,B: nat > real,C2: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_real @ ( B @ A ) @ ( groups129246275422532515t_real @ C2 @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups129246275422532515t_real @ C2 @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7394_prod_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: complex > rat,C2: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups225925009352817453ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_rat @ ( B @ A ) @ ( groups225925009352817453ex_rat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups225925009352817453ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups225925009352817453ex_rat @ C2 @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7395_prod_Odelta__remove,axiom,
! [S2: set_real,A: real,B: real > rat,C2: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times_rat @ ( B @ A ) @ ( groups4061424788464935467al_rat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups4061424788464935467al_rat @ C2 @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_7396_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb3: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb3 @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb3 @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb3 @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb3 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_7397_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F5: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F5 @ A5 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_7398_pi__half__less__two,axiom,
ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_less_two
thf(fact_7399_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7400_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7401_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7402_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7403_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7404_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7405_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7406_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7407_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7408_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7409_prod__diff1,axiom,
! [A4: set_complex,F: complex > complex,A: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( F @ A )
!= zero_zero_complex )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups3708469109370488835omplex @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7410_prod__diff1,axiom,
! [A4: set_complex,F: complex > rat,A: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( F @ A )
!= zero_zero_rat )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups225925009352817453ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( divide_divide_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups225925009352817453ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7411_prod__diff1,axiom,
! [A4: set_real,F: real > complex,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( ( F @ A )
!= zero_zero_complex )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups713298508707869441omplex @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7412_prod__diff1,axiom,
! [A4: set_real,F: real > rat,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( ( F @ A )
!= zero_zero_rat )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups4061424788464935467al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( divide_divide_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups4061424788464935467al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7413_prod__diff1,axiom,
! [A4: set_int,F: int > complex,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( ( F @ A )
!= zero_zero_complex )
=> ( ( ( member_int @ A @ A4 )
=> ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A4 )
=> ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups7440179247065528705omplex @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7414_prod__diff1,axiom,
! [A4: set_int,F: int > rat,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( ( F @ A )
!= zero_zero_rat )
=> ( ( ( member_int @ A @ A4 )
=> ( ( groups1072433553688619179nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( divide_divide_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A4 )
=> ( ( groups1072433553688619179nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7415_prod__diff1,axiom,
! [A4: set_nat,F: nat > complex,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( F @ A )
!= zero_zero_complex )
=> ( ( ( member_nat @ A @ A4 )
=> ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups6464643781859351333omplex @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7416_prod__diff1,axiom,
! [A4: set_nat,F: nat > rat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( F @ A )
!= zero_zero_rat )
=> ( ( ( member_nat @ A @ A4 )
=> ( ( groups73079841787564623at_rat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( divide_divide_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( groups73079841787564623at_rat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups73079841787564623at_rat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7417_prod__diff1,axiom,
! [A4: set_complex,F: complex > nat,A: complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( F @ A )
!= zero_zero_nat )
=> ( ( ( member_complex @ A @ A4 )
=> ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( divide_divide_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_complex @ A @ A4 )
=> ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7418_prod__diff1,axiom,
! [A4: set_real,F: real > nat,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( ( F @ A )
!= zero_zero_nat )
=> ( ( ( member_real @ A @ A4 )
=> ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( divide_divide_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( F @ A ) ) ) )
& ( ~ ( member_real @ A @ A4 )
=> ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_7419_log__of__power__less,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_7420_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_7421_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7422_pochhammer__Suc__prod,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7423_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7424_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7425_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_gt_zero
thf(fact_7426_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A5: rat,N2: nat] :
( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7427_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N2: nat] :
( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7428_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A5: int,N2: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7429_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N2: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7430_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% m2pi_less_pi
thf(fact_7431_log__of__power__le,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_7432_prod_Oin__pairs,axiom,
! [G: nat > complex,M2: nat,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7433_prod_Oin__pairs,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7434_prod_Oin__pairs,axiom,
! [G: nat > rat,M2: nat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7435_prod_Oin__pairs,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7436_prod_Oin__pairs,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7437_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_7438_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7439_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_7440_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7441_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_7442_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7443_pochhammer__Suc__prod__rev,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7444_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7445_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7446_minus__pi__half__less__zero,axiom,
ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% minus_pi_half_less_zero
thf(fact_7447_less__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% less_log2_of_power
thf(fact_7448_le__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_7449_log2__of__power__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_7450_log2__of__power__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_7451_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K2: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
=> ( ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_7452_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% ceiling_log2_div2
thf(fact_7453_ceiling__log__eq__powr__iff,axiom,
! [X: real,B: real,K2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ K2 ) @ one_one_int ) )
= ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ X )
& ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_7454_geometric__deriv__sums,axiom,
! [Z: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7455_geometric__deriv__sums,axiom,
! [Z: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7456_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_7457_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_7458_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L ) )
=> ( ( ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K2 @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K2 @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_7459_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_7460_powr__eq__0__iff,axiom,
! [W2: real,Z: real] :
( ( ( powr_real @ W2 @ Z )
= zero_zero_real )
= ( W2 = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_7461_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_7462_sums__zero,axiom,
( sums_complex
@ ^ [N2: nat] : zero_zero_complex
@ zero_zero_complex ) ).
% sums_zero
thf(fact_7463_sums__zero,axiom,
( sums_real
@ ^ [N2: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_7464_sums__zero,axiom,
( sums_nat
@ ^ [N2: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_7465_sums__zero,axiom,
( sums_int
@ ^ [N2: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_7466_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_7467_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_7468_floor__zero,axiom,
( ( archim3151403230148437115or_rat @ zero_zero_rat )
= zero_zero_int ) ).
% floor_zero
thf(fact_7469_powr__gt__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_7470_powr__less__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_7471_prod__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7472_prod__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups861055069439313189ex_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7473_prod__eq__1__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups4077766827762148844at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7474_prod__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups708209901874060359at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7475_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_7476_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_7477_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_7478_powr__le__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_7479_prod__pos__nat__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_7480_prod__pos__nat__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_7481_prod__pos__nat__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_7482_prod__pos__nat__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_7483_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_7484_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_7485_zero__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_floor
thf(fact_7486_zero__le__floor,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% zero_le_floor
thf(fact_7487_floor__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_7488_floor__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
= ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% floor_less_zero
thf(fact_7489_numeral__le__floor,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% numeral_le_floor
thf(fact_7490_numeral__le__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% numeral_le_floor
thf(fact_7491_zero__less__floor,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% zero_less_floor
thf(fact_7492_zero__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% zero_less_floor
thf(fact_7493_floor__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_le_zero
thf(fact_7494_floor__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_le_zero
thf(fact_7495_floor__less__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% floor_less_numeral
thf(fact_7496_floor__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% floor_less_numeral
thf(fact_7497_one__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% one_le_floor
thf(fact_7498_one__le__floor,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% one_le_floor
thf(fact_7499_floor__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_less_one
thf(fact_7500_floor__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_less_one
thf(fact_7501_log__powr__cancel,axiom,
! [A: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr_real @ A @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_7502_powr__log__cancel,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ A @ ( log @ A @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_7503_powser__sums__zero__iff,axiom,
! [A: nat > complex,X: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7504_powser__sums__zero__iff,axiom,
! [A: nat > real,X: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7505_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_7506_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_7507_numeral__less__floor,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% numeral_less_floor
thf(fact_7508_numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_less_floor
thf(fact_7509_floor__le__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% floor_le_numeral
thf(fact_7510_floor__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% floor_le_numeral
thf(fact_7511_one__less__floor,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% one_less_floor
thf(fact_7512_one__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% one_less_floor
thf(fact_7513_floor__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% floor_le_one
thf(fact_7514_floor__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% floor_le_one
thf(fact_7515_neg__numeral__le__floor,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% neg_numeral_le_floor
thf(fact_7516_neg__numeral__le__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_le_floor
thf(fact_7517_floor__less__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_7518_floor__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_7519_neg__numeral__less__floor,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% neg_numeral_less_floor
thf(fact_7520_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_less_floor
thf(fact_7521_floor__le__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% floor_le_neg_numeral
thf(fact_7522_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% floor_le_neg_numeral
thf(fact_7523_int__prod,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_prod
thf(fact_7524_int__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_prod
thf(fact_7525_sums__0,axiom,
! [F: nat > complex] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_complex )
=> ( sums_complex @ F @ zero_zero_complex ) ) ).
% sums_0
thf(fact_7526_sums__0,axiom,
! [F: nat > real] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_7527_sums__0,axiom,
! [F: nat > nat] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_7528_sums__0,axiom,
! [F: nat > int] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_7529_sums__le,axiom,
! [F: nat > real,G: nat > real,S: real,T: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_real @ F @ S )
=> ( ( sums_real @ G @ T )
=> ( ord_less_eq_real @ S @ T ) ) ) ) ).
% sums_le
thf(fact_7530_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_nat @ F @ S )
=> ( ( sums_nat @ G @ T )
=> ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% sums_le
thf(fact_7531_sums__le,axiom,
! [F: nat > int,G: nat > int,S: int,T: int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_int @ F @ S )
=> ( ( sums_int @ G @ T )
=> ( ord_less_eq_int @ S @ T ) ) ) ) ).
% sums_le
thf(fact_7532_sums__single,axiom,
! [I2: nat,F: nat > complex] :
( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( F @ I2 ) ) ).
% sums_single
thf(fact_7533_sums__single,axiom,
! [I2: nat,F: nat > real] :
( sums_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( F @ I2 ) ) ).
% sums_single
thf(fact_7534_sums__single,axiom,
! [I2: nat,F: nat > nat] :
( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( F @ I2 ) ) ).
% sums_single
thf(fact_7535_sums__single,axiom,
! [I2: nat,F: nat > int] :
( sums_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( F @ I2 ) ) ).
% sums_single
thf(fact_7536_sums__add,axiom,
! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
( ( sums_nat @ F @ A )
=> ( ( sums_nat @ G @ B )
=> ( sums_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_nat @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7537_sums__add,axiom,
! [F: nat > int,A: int,G: nat > int,B: int] :
( ( sums_int @ F @ A )
=> ( ( sums_int @ G @ B )
=> ( sums_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_int @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7538_sums__add,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_real @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7539_floor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% floor_mono
thf(fact_7540_floor__mono,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% floor_mono
thf(fact_7541_of__int__floor__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_7542_of__int__floor__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_7543_floor__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_7544_floor__less__cancel,axiom,
! [X: rat,Y: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_7545_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_7546_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_7547_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel
thf(fact_7548_powr__less__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_less_mono
thf(fact_7549_sums__mult__iff,axiom,
! [C2: complex,F: nat > complex,D: complex] :
( ( C2 != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C2 @ ( F @ N2 ) )
@ ( times_times_complex @ C2 @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7550_sums__mult__iff,axiom,
! [C2: real,F: nat > real,D: real] :
( ( C2 != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C2 @ ( F @ N2 ) )
@ ( times_times_real @ C2 @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7551_sums__mult2__iff,axiom,
! [C2: complex,F: nat > complex,D: complex] :
( ( C2 != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C2 )
@ ( times_times_complex @ D @ C2 ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7552_sums__mult2__iff,axiom,
! [C2: real,F: nat > real,D: real] :
( ( C2 != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C2 )
@ ( times_times_real @ D @ C2 ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7553_le__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_7554_le__floor__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_7555_floor__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% floor_less_iff
thf(fact_7556_floor__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% floor_less_iff
thf(fact_7557_powr__mono2_H,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_7558_powr__less__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_7559_le__floor__add,axiom,
! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% le_floor_add
thf(fact_7560_le__floor__add,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% le_floor_add
thf(fact_7561_powr__inj,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= ( powr_real @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_7562_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_7563_int__add__floor,axiom,
! [Z: int,X: real] :
( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).
% int_add_floor
thf(fact_7564_int__add__floor,axiom,
! [Z: int,X: rat] :
( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).
% int_add_floor
thf(fact_7565_floor__add__int,axiom,
! [X: real,Z: int] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% floor_add_int
thf(fact_7566_floor__add__int,axiom,
! [X: rat,Z: int] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% floor_add_int
thf(fact_7567_sums__mult__D,axiom,
! [C2: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C2 @ ( F @ N2 ) )
@ A )
=> ( ( C2 != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C2 ) ) ) ) ).
% sums_mult_D
thf(fact_7568_sums__mult__D,axiom,
! [C2: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C2 @ ( F @ N2 ) )
@ A )
=> ( ( C2 != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% sums_mult_D
thf(fact_7569_sums__Suc__imp,axiom,
! [F: nat > complex,S: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S )
=> ( sums_complex @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_7570_sums__Suc__imp,axiom,
! [F: nat > real,S: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S )
=> ( sums_real @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_7571_sums__Suc__iff,axiom,
! [F: nat > real,S: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S )
= ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_7572_sums__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( sums_nat
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L )
=> ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7573_sums__Suc,axiom,
! [F: nat > int,L: int] :
( ( sums_int
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L )
=> ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7574_sums__Suc,axiom,
! [F: nat > real,L: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L )
=> ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7575_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > complex,S: complex] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( F @ I3 )
= zero_zero_complex ) )
=> ( ( sums_complex
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S )
= ( sums_complex @ F @ S ) ) ) ).
% sums_zero_iff_shift
thf(fact_7576_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S: real] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( F @ I3 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S )
= ( sums_real @ F @ S ) ) ) ).
% sums_zero_iff_shift
thf(fact_7577_powr__add,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% powr_add
thf(fact_7578_sums__finite,axiom,
! [N6: set_nat,F: nat > complex] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_complex ) )
=> ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_7579_sums__finite,axiom,
! [N6: set_nat,F: nat > int] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_int ) )
=> ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_7580_sums__finite,axiom,
! [N6: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_nat ) )
=> ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_7581_sums__finite,axiom,
! [N6: set_nat,F: nat > real] :
( ( finite_finite_nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N6 )
=> ( ( F @ N3 )
= zero_zero_real ) )
=> ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% sums_finite
thf(fact_7582_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > complex] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7583_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7584_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7585_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7586_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( groups2073611262835488442omplex @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7587_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7588_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7589_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7590_prod__int__eq,axiom,
! [I2: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_7591_one__add__floor,axiom,
! [X: real] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% one_add_floor
thf(fact_7592_one__add__floor,axiom,
! [X: rat] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% one_add_floor
thf(fact_7593_floor__log__eq__powr__iff,axiom,
! [X: real,B: real,K2: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
= K2 )
= ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K2 ) ) @ X )
& ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_7594_powser__sums__if,axiom,
! [M2: nat,Z: complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
@ ( power_power_complex @ Z @ M2 ) ) ).
% powser_sums_if
thf(fact_7595_powser__sums__if,axiom,
! [M2: nat,Z: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
@ ( power_power_real @ Z @ M2 ) ) ).
% powser_sums_if
thf(fact_7596_powser__sums__if,axiom,
! [M2: nat,Z: int] :
( sums_int
@ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
@ ( power_power_int @ Z @ M2 ) ) ).
% powser_sums_if
thf(fact_7597_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7598_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7599_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_7600_powr__less__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
= ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_7601_less__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
= ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_7602_log__less__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B @ X ) @ Y )
= ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_7603_less__log__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ ( log @ B @ X ) )
= ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_7604_sums__iff__shift,axiom,
! [F: nat > real,N: nat,S: real] :
( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S )
= ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_iff_shift
thf(fact_7605_sums__split__initial__segment,axiom,
! [F: nat > real,S: real,N: nat] :
( ( sums_real @ F @ S )
=> ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_7606_sums__iff__shift_H,axiom,
! [F: nat > real,N: nat,S: real] :
( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
= ( sums_real @ F @ S ) ) ).
% sums_iff_shift'
thf(fact_7607_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_7608_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_7609_sums__If__finite__set_H,axiom,
! [G: nat > real,S2: real,A4: set_nat,S5: real,F: nat > real] :
( ( sums_real @ G @ S2 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( S5
= ( plus_plus_real @ S2
@ ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ A4 ) ) )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
@ S5 ) ) ) ) ).
% sums_If_finite_set'
thf(fact_7610_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_7611_prod__int__plus__eq,axiom,
! [I2: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_7612_floor__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= Z ) ) ) ).
% floor_unique
thf(fact_7613_floor__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
=> ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
=> ( ( archim3151403230148437115or_rat @ X )
= Z ) ) ) ).
% floor_unique
thf(fact_7614_floor__eq__iff,axiom,
! [X: real,A: int] :
( ( ( archim6058952711729229775r_real @ X )
= A )
= ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% floor_eq_iff
thf(fact_7615_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim3151403230148437115or_rat @ X )
= A )
= ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% floor_eq_iff
thf(fact_7616_floor__split,axiom,
! [P2: int > $o,T: real] :
( ( P2 @ ( archim6058952711729229775r_real @ T ) )
= ( ! [I4: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
& ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% floor_split
thf(fact_7617_floor__split,axiom,
! [P2: int > $o,T: rat] :
( ( P2 @ ( archim3151403230148437115or_rat @ T ) )
= ( ! [I4: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
& ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% floor_split
thf(fact_7618_le__mult__floor,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% le_mult_floor
thf(fact_7619_le__mult__floor,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% le_mult_floor
thf(fact_7620_less__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% less_floor_iff
thf(fact_7621_less__floor__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% less_floor_iff
thf(fact_7622_floor__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% floor_le_iff
thf(fact_7623_floor__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% floor_le_iff
thf(fact_7624_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_7625_binomial__mono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_mono
thf(fact_7626_binomial__antimono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K2 )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_antimono
thf(fact_7627_binomial__maximum,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_7628_floor__correct,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_7629_floor__correct,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_7630_powr__neg__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X ) ) ) ).
% powr_neg_one
thf(fact_7631_le__log__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_7632_log__le__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_7633_le__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
= ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_7634_powr__le__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
= ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_7635_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_7636_ln__prod,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I6 ) )
= ( groups8097168146408367636l_real
@ ^ [X5: real] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7637_ln__prod,axiom,
! [I6: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ I6 )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups3619160379726066777t_real @ F @ I6 ) )
= ( groups5107569545109728110t_real
@ ^ [X5: set_nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7638_ln__prod,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I6 ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7639_ln__prod,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I6 ) )
= ( groups5808333547571424918x_real
@ ^ [X5: complex] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7640_ln__prod,axiom,
! [I6: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ I6 )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups6036352826371341000t_real @ F @ I6 ) )
= ( groups4567486121110086003t_real
@ ^ [X5: product_prod_nat_nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7641_ln__prod,axiom,
! [I6: set_nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I6 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_7642_floor__divide__lower,axiom,
! [Q2: real,P: real] :
( ( ord_less_real @ zero_zero_real @ Q2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P @ Q2 ) ) ) @ Q2 ) @ P ) ) ).
% floor_divide_lower
thf(fact_7643_floor__divide__lower,axiom,
! [Q2: rat,P: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P @ Q2 ) ) ) @ Q2 ) @ P ) ) ).
% floor_divide_lower
thf(fact_7644_binomial__less__binomial__Suc,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ K2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_7645_binomial__strict__mono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_7646_binomial__strict__antimono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_7647_ln__powr__bound,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% ln_powr_bound
thf(fact_7648_ln__powr__bound2,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_7649_log__add__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_7650_add__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_7651_minus__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_7652_floor__divide__upper,axiom,
! [Q2: real,P: real] :
( ( ord_less_real @ zero_zero_real @ Q2 )
=> ( ord_less_real @ P @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% floor_divide_upper
thf(fact_7653_floor__divide__upper,axiom,
! [Q2: rat,P: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q2 )
=> ( ord_less_rat @ P @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% floor_divide_upper
thf(fact_7654_geometric__sums,axiom,
! [C2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C2 ) @ one_one_real )
=> ( sums_real @ ( power_power_real @ C2 ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C2 ) ) ) ) ).
% geometric_sums
thf(fact_7655_geometric__sums,axiom,
! [C2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C2 ) @ one_one_real )
=> ( sums_complex @ ( power_power_complex @ C2 ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C2 ) ) ) ) ).
% geometric_sums
thf(fact_7656_round__def,axiom,
( archim8280529875227126926d_real
= ( ^ [X5: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X5 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_7657_round__def,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X5: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X5 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_7658_log__minus__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_7659_and__int_Opinduct,axiom,
! [A0: int,A1: int,P2: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [K: int,L4: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L4 ) )
=> ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ K @ L4 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_7660_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M3: nat,N2: nat] :
( if_nat
@ ( ( M3 = zero_zero_nat )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_7661_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_7662_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M3: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_7663_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% floor_log2_div2
thf(fact_7664_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
=> ( ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_7665_zero__less__binomial__iff,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) )
= ( ord_less_eq_nat @ K2 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_7666_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_7667_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_7668_binomial__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( binomial @ N @ K2 )
= zero_zero_nat )
= ( ord_less_nat @ N @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_7669_binomial__0__Suc,axiom,
! [K2: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_7670_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_7671_binomial__eq__0,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( binomial @ N @ K2 )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_7672_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_7673_choose__mult__lemma,axiom,
! [M2: nat,R2: nat,K2: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus_nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K2 ) @ K2 ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_7674_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_7675_zero__less__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_7676_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
thf(fact_7677_choose__mult,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
= ( times_times_nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_7678_binomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_7679_binomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_7680_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_7681_choose__reduce__nat,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( binomial @ N @ K2 )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_7682_times__binomial__minus1__eq,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( times_times_nat @ K2 @ ( binomial @ N @ K2 ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_7683_binomial__addition__formula,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_7684_upto_Opinduct,axiom,
! [A0: int,A1: int,P2: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I3 @ J2 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
=> ( P2 @ I3 @ J2 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_7685_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] :
( if_complex
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7686_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7687_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7688_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( if_real
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7689_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7690_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7691_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7692_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7693_arcosh__def,axiom,
( arcosh_real
= ( ^ [X5: real] : ( ln_ln_real @ ( plus_plus_real @ X5 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_7694_round__altdef,axiom,
( archim8280529875227126926d_real
= ( ^ [X5: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X5 ) ) @ ( archim7802044766580827645g_real @ X5 ) @ ( archim6058952711729229775r_real @ X5 ) ) ) ) ).
% round_altdef
thf(fact_7695_round__altdef,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X5: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X5 ) ) @ ( archim2889992004027027881ng_rat @ X5 ) @ ( archim3151403230148437115or_rat @ X5 ) ) ) ) ).
% round_altdef
thf(fact_7696_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_7697_atMost__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_atMost_int @ X )
= ( set_ord_atMost_int @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_7698_atMost__iff,axiom,
! [I2: real,K2: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K2 ) )
= ( ord_less_eq_real @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7699_atMost__iff,axiom,
! [I2: set_nat,K2: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K2 ) )
= ( ord_less_eq_set_nat @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7700_atMost__iff,axiom,
! [I2: set_int,K2: set_int] :
( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K2 ) )
= ( ord_less_eq_set_int @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7701_atMost__iff,axiom,
! [I2: rat,K2: rat] :
( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K2 ) )
= ( ord_less_eq_rat @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7702_atMost__iff,axiom,
! [I2: num,K2: num] :
( ( member_num @ I2 @ ( set_ord_atMost_num @ K2 ) )
= ( ord_less_eq_num @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7703_atMost__iff,axiom,
! [I2: nat,K2: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7704_atMost__iff,axiom,
! [I2: int,K2: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K2 ) )
= ( ord_less_eq_int @ I2 @ K2 ) ) ).
% atMost_iff
thf(fact_7705_finite__atMost,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K2 ) ) ).
% finite_atMost
thf(fact_7706_atMost__subset__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7707_atMost__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7708_atMost__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7709_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7710_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7711_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_7712_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_7713_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_7714_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_7715_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_add
thf(fact_7716_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_add
thf(fact_7717_frac__of__int,axiom,
! [Z: int] :
( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
= zero_zero_real ) ).
% frac_of_int
thf(fact_7718_frac__of__int,axiom,
! [Z: int] :
( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
= zero_zero_rat ) ).
% frac_of_int
thf(fact_7719_Icc__subset__Iic__iff,axiom,
! [L: set_int,H2: set_int,H3: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
= ( ~ ( ord_less_eq_set_int @ L @ H2 )
| ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7720_Icc__subset__Iic__iff,axiom,
! [L: rat,H2: rat,H3: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
= ( ~ ( ord_less_eq_rat @ L @ H2 )
| ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7721_Icc__subset__Iic__iff,axiom,
! [L: num,H2: num,H3: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
= ( ~ ( ord_less_eq_num @ L @ H2 )
| ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7722_Icc__subset__Iic__iff,axiom,
! [L: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7723_Icc__subset__Iic__iff,axiom,
! [L: int,H2: int,H3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
= ( ~ ( ord_less_eq_int @ L @ H2 )
| ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7724_Icc__subset__Iic__iff,axiom,
! [L: real,H2: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L @ H2 )
| ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_7725_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_7726_sum_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_7727_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_7728_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_7729_prod_OatMost__Suc,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7730_prod_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7731_prod_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7732_prod_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7733_prod_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7734_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_7735_not__empty__eq__Iic__eq__empty,axiom,
! [H2: real] :
( bot_bot_set_real
!= ( set_ord_atMost_real @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7736_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7737_not__empty__eq__Iic__eq__empty,axiom,
! [H2: int] :
( bot_bot_set_int
!= ( set_ord_atMost_int @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7738_infinite__Iic,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% infinite_Iic
thf(fact_7739_not__Iic__eq__Icc,axiom,
! [H3: int,L: int,H2: int] :
( ( set_ord_atMost_int @ H3 )
!= ( set_or1266510415728281911st_int @ L @ H2 ) ) ).
% not_Iic_eq_Icc
thf(fact_7740_not__Iic__eq__Icc,axiom,
! [H3: real,L: real,H2: real] :
( ( set_ord_atMost_real @ H3 )
!= ( set_or1222579329274155063t_real @ L @ H2 ) ) ).
% not_Iic_eq_Icc
thf(fact_7741_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_eq_real @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7742_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] : ( ord_less_eq_set_nat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7743_atMost__def,axiom,
( set_or58775011639299419et_int
= ( ^ [U2: set_int] :
( collect_set_int
@ ^ [X5: set_int] : ( ord_less_eq_set_int @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7744_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X5: rat] : ( ord_less_eq_rat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7745_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X5: num] : ( ord_less_eq_num @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7746_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_eq_nat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7747_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_eq_int @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7748_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_7749_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_7750_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost_nat @ ( suc @ K2 ) )
= ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_7751_not__Iic__le__Icc,axiom,
! [H2: int,L3: int,H3: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_7752_not__Iic__le__Icc,axiom,
! [H2: real,L3: real,H3: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_7753_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S7: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_7754_frac__ge__0,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% frac_ge_0
thf(fact_7755_frac__ge__0,axiom,
! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% frac_ge_0
thf(fact_7756_frac__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% frac_lt_1
thf(fact_7757_frac__lt__1,axiom,
! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% frac_lt_1
thf(fact_7758_frac__1__eq,axiom,
! [X: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( archim2898591450579166408c_real @ X ) ) ).
% frac_1_eq
thf(fact_7759_frac__1__eq,axiom,
! [X: rat] :
( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( archimedean_frac_rat @ X ) ) ).
% frac_1_eq
thf(fact_7760_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( numeral_numeral_nat @ K2 ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_7761_Iic__subset__Iio__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_7762_Iic__subset__Iio__iff,axiom,
! [A: num,B: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
= ( ord_less_num @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_7763_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_7764_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_7765_Iic__subset__Iio__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_7766_norm__less__p1,axiom,
! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% norm_less_p1
thf(fact_7767_norm__less__p1,axiom,
! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% norm_less_p1
thf(fact_7768_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7769_sum_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7770_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7771_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7772_sum__telescope,axiom,
! [F: nat > int,I2: nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_7773_sum__telescope,axiom,
! [F: nat > rat,I2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_7774_sum__telescope,axiom,
! [F: nat > real,I2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_7775_polyfun__eq__coeffs,axiom,
! [C2: nat > complex,N: nat,D: nat > complex] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( D @ I4 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_7776_polyfun__eq__coeffs,axiom,
! [C2: nat > real,N: nat,D: nat > real] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( D @ I4 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_7777_prod_OatMost__Suc__shift,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_complex @ ( G @ zero_zero_nat )
@ ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7778_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7779_prod_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7780_prod_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7781_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7782_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_7783_sum_Onested__swap_H,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_7784_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I4: nat] : ( groups705719431365010083at_int @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_7785_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_7786_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_7787_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_7788_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_7789_frac__eq,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= X )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% frac_eq
thf(fact_7790_frac__eq,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= X )
= ( ( ord_less_eq_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% frac_eq
thf(fact_7791_polyfun__eq__0,axiom,
! [C2: nat > complex,N: nat] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C2 @ I4 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_7792_polyfun__eq__0,axiom,
! [C2: nat > real,N: nat] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C2 @ I4 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_7793_zero__polynom__imp__zero__coeffs,axiom,
! [C2: nat > complex,N: nat,K2: nat] :
( ! [W: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ W @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( C2 @ K2 )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_7794_zero__polynom__imp__zero__coeffs,axiom,
! [C2: nat > real,N: nat,K2: nat] :
( ! [W: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ W @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( C2 @ K2 )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_7795_frac__add,axiom,
! [X: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% frac_add
thf(fact_7796_frac__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% frac_add
thf(fact_7797_sum_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_7798_sum_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_7799_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_7800_sum_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_7801_sum__up__index__split,axiom,
! [F: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_7802_sum__up__index__split,axiom,
! [F: nat > rat,M2: nat,N: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_7803_sum__up__index__split,axiom,
! [F: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_7804_sum__up__index__split,axiom,
! [F: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_7805_prod_OatMost__shift,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( G @ zero_zero_nat )
@ ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_7806_prod_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_7807_prod_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_7808_prod_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_7809_prod_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_7810_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_7811_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_7812_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_7813_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_7814_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_7815_sum__choose__diagonal,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M2 @ K3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( suc @ N ) @ M2 ) ) ) ).
% sum_choose_diagonal
thf(fact_7816_vandermonde,axiom,
! [M2: nat,N: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M2 @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_7817_sum__gp__basic,axiom,
! [X: complex,N: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7818_sum__gp__basic,axiom,
! [X: rat,N: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7819_sum__gp__basic,axiom,
! [X: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7820_sum__gp__basic,axiom,
! [X: real,N: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7821_polyfun__finite__roots,axiom,
! [C2: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( C2 @ I4 )
!= zero_zero_complex ) ) ) ) ).
% polyfun_finite_roots
thf(fact_7822_polyfun__finite__roots,axiom,
! [C2: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( C2 @ I4 )
!= zero_zero_real ) ) ) ) ).
% polyfun_finite_roots
thf(fact_7823_polyfun__roots__finite,axiom,
! [C2: nat > complex,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_7824_polyfun__roots__finite,axiom,
! [C2: nat > real,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_7825_polyfun__linear__factor__root,axiom,
! [C2: nat > complex,A: complex,N: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ~ ! [B3: nat > complex] :
~ ! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_7826_polyfun__linear__factor__root,axiom,
! [C2: nat > rat,A: rat,N: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat )
=> ~ ! [B3: nat > rat] :
~ ! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_7827_polyfun__linear__factor__root,axiom,
! [C2: nat > int,A: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C2 @ I4 ) @ ( power_power_int @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B3: nat > int] :
~ ! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C2 @ I4 ) @ ( power_power_int @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_7828_polyfun__linear__factor__root,axiom,
! [C2: nat > real,A: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B3: nat > real] :
~ ! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_7829_polyfun__linear__factor,axiom,
! [C2: nat > complex,N: nat,A: complex] :
? [B3: nat > complex] :
! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_complex
@ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_7830_polyfun__linear__factor,axiom,
! [C2: nat > rat,N: nat,A: rat] :
? [B3: nat > rat] :
! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat
@ ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C2 @ I4 ) @ ( power_power_rat @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_7831_polyfun__linear__factor,axiom,
! [C2: nat > int,N: nat,A: int] :
? [B3: nat > int] :
! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C2 @ I4 ) @ ( power_power_int @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C2 @ I4 ) @ ( power_power_int @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_7832_polyfun__linear__factor,axiom,
! [C2: nat > real,N: nat,A: real] :
? [B3: nat > real] :
! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_7833_sum__power__shift,axiom,
! [M2: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7834_sum__power__shift,axiom,
! [M2: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7835_sum__power__shift,axiom,
! [M2: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7836_sum__power__shift,axiom,
! [M2: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7837_sum_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_7838_sum_Otriangle__reindex,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_7839_prod_Otriangle__reindex,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_7840_prod_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_7841_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_7842_sum_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7843_sum_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7844_sum_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7845_sum_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7846_polynomial__product,axiom,
! [M2: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
( ! [I3: nat] :
( ( ord_less_nat @ M2 @ I3 )
=> ( ( A @ I3 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_complex @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_7847_polynomial__product,axiom,
! [M2: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
( ! [I3: nat] :
( ( ord_less_nat @ M2 @ I3 )
=> ( ( A @ I3 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2906978787729119204at_rat
@ ^ [R5: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_rat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_7848_polynomial__product,axiom,
! [M2: nat,A: nat > int,N: nat,B: nat > int,X: int] :
( ! [I3: nat] :
( ( ord_less_nat @ M2 @ I3 )
=> ( ( A @ I3 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [R5: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_int @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_7849_polynomial__product,axiom,
! [M2: nat,A: nat > real,N: nat,B: nat > real,X: real] :
( ! [I3: nat] :
( ( ord_less_nat @ M2 @ I3 )
=> ( ( A @ I3 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [R5: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_real @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_7850_prod_Oin__pairs__0,axiom,
! [G: nat > complex,N: nat] :
( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7851_prod_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7852_prod_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7853_prod_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7854_prod_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7855_polyfun__eq__const,axiom,
! [C2: nat > complex,N: nat,K2: complex] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C2 @ I4 ) @ ( power_power_complex @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= K2 ) )
= ( ( ( C2 @ zero_zero_nat )
= K2 )
& ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C2 @ X5 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_7856_polyfun__eq__const,axiom,
! [C2: nat > real,N: nat,K2: real] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C2 @ I4 ) @ ( power_power_real @ X5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= K2 ) )
= ( ( ( C2 @ zero_zero_nat )
= K2 )
& ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C2 @ X5 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_7857_binomial__ring,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7858_binomial__ring,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7859_binomial__ring,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7860_binomial__ring,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7861_binomial__ring,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7862_pochhammer__binomial__sum,axiom,
! [A: complex,B: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( comm_s2602460028002588243omplex @ A @ K3 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7863_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7864_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7865_pochhammer__binomial__sum,axiom,
! [A: real,B: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7866_polynomial__product__nat,axiom,
! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ M2 @ I3 )
=> ( ( A @ I3 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_7867_floor__add,axiom,
! [X: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_7868_floor__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_7869_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7870_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7871_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7872_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7873_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7874_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7875_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7876_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7877_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7878_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7879_root__polyfun,axiom,
! [N: nat,Z: complex,A: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_complex @ Z @ N )
= A )
= ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ).
% root_polyfun
thf(fact_7880_root__polyfun,axiom,
! [N: nat,Z: int,A: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_int @ Z @ N )
= A )
= ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ) ).
% root_polyfun
thf(fact_7881_root__polyfun,axiom,
! [N: nat,Z: rat,A: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_rat @ Z @ N )
= A )
= ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ) ).
% root_polyfun
thf(fact_7882_root__polyfun,axiom,
! [N: nat,Z: real,A: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z @ N )
= A )
= ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_7883_sum__gp0,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_7884_sum__gp0,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_7885_sum__gp0,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_7886_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_linear_sum
thf(fact_7887_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_7888_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_7889_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_linear_sum
thf(fact_7890_polyfun__diff__alt,axiom,
! [N: nat,A: nat > complex,X: complex,Y: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_7891_polyfun__diff__alt,axiom,
! [N: nat,A: nat > rat,X: rat,Y: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_7892_polyfun__diff__alt,axiom,
! [N: nat,A: nat > int,X: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_7893_polyfun__diff__alt,axiom,
! [N: nat,A: nat > real,X: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_7894_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_7895_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_sum
thf(fact_7896_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_7897_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_7898_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_sum
thf(fact_7899_cot__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% cot_less_zero
thf(fact_7900_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_7901_of__nat__id,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] : N2 ) ) ).
% of_nat_id
thf(fact_7902_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_7903_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_7904_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_less_exp_iff
thf(fact_7905_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_7906_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln_ln_real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_7907_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln_ln_real @ X ) )
= X )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% exp_ln_iff
thf(fact_7908_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_7909_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( exp_real @ X4 )
= Y ) ) ).
% exp_total
thf(fact_7910_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_gt_zero
thf(fact_7911_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_7912_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% exp_gt_one
thf(fact_7913_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_7914_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_7915_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_7916_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_7917_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_7918_sum__pos__lt__pair,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real @ F )
=> ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K2 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K2 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_7919_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_7920_modulo__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) )
& ( ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_7921_powr__int,axiom,
! [X: real,I2: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( power_power_real @ X @ ( nat2 @ I2 ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_7922_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_7923_nat__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_nat @ K2 ) ) ).
% nat_numeral
thf(fact_7924_nat__of__bool,axiom,
! [P2: $o] :
( ( nat2 @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% nat_of_bool
thf(fact_7925_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_7926_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_7927_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_7928_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_7929_nat__neg__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_7930_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_7931_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_7932_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K2: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K2 )
= ( ( dvd_dvd_int @ L @ K2 )
& ( ( R2 = zero_zero_int )
=> ( K2 = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_7933_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K2 )
= ( ( dvd_dvd_int @ L @ K2 )
& ( ( R2 = zero_zero_int )
=> ( K2 = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_7934_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K2 ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_7935_dvd__mult__sgn__iff,axiom,
! [L: int,K2: int,R2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K2 @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_7936_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_7937_diff__nat__numeral,axiom,
! [V: num,V3: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_7938_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_7939_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_7940_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_7941_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_7942_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_7943_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_7944_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_7945_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_7946_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_7947_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_7948_fact__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_7949_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_7950_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_7951_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_7952_int__sgnE,axiom,
! [K2: int] :
~ ! [N3: nat,L4: int] :
( K2
!= ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_sgnE
thf(fact_7953_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_7954_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% all_nat
thf(fact_7955_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
& ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% ex_nat
thf(fact_7956_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_7957_dvd__fact,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7958_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_7959_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_7960_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_7961_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_7962_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_7963_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_7964_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_7965_fact__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_7966_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_7967_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_7968_binomial__fact__lemma,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_7969_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_7970_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_7971_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_7972_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_7973_split__nat,axiom,
! [P2: nat > $o,I2: int] :
( ( P2 @ ( nat2 @ I2 ) )
= ( ! [N2: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P2 @ N2 ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_7974_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_7975_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_7976_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_7977_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ord_less_eq_int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_7978_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_7979_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_7980_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_7981_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_7982_le__nat__floor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_7983_fact__eq__fact__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1408675320244567234ct_nat @ M2 )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_7984_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_7985_binomial__altdef__nat,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_7986_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_7987_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_7988_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_7989_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_7990_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_7991_diff__nat__eq__if,axiom,
! [Z7: int,Z: int] :
( ( ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_7992_summable__power__series,axiom,
! [F: nat > real,Z: real] :
( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
=> ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( summable_real
@ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_7993_nat__dvd__iff,axiom,
! [Z: int,M2: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_7994_Maclaurin__lemma,axiom,
! [H2: real,F: real > real,J: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ? [B8: real] :
( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_7995_divide__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_7996_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_7997_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_7998_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
& ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_7999_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( sin_real @ X ) ) ).
% sin_paired
thf(fact_8000_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_8001_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_8002_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
thf(fact_8003_sgn__power__injE,axiom,
! [A: real,N: nat,X: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_8004_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_8005_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_8006_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_8007_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_8008_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
thf(fact_8009_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_8010_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_8011_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_8012_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_lt_zero
thf(fact_8013_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_le_zero
thf(fact_8014_sin__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_less_zero
thf(fact_8015_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_8016_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_8017_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_8018_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_8019_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_8020_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ T6 )
& ( ord_less_real @ T6 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_8021_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y6: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y6 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y6 )
& ( ord_less_eq_real @ Y6 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_8022_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_8023_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_8024_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y6: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y6 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y6 )
& ( ord_less_real @ Y6 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_8025_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_arctan_iff
thf(fact_8026_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% arctan_less_zero_iff
thf(fact_8027_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_8028_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_8029_dvd__nat__abs__iff,axiom,
! [N: nat,K2: int] :
( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K2 ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ).
% dvd_nat_abs_iff
thf(fact_8030_nat__abs__dvd__iff,axiom,
! [K2: int,N: nat] :
( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ N )
= ( dvd_dvd_int @ K2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_8031_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_8032_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_8033_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_8034_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M2 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_8035_infinite__int__iff__unbounded__le,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M3: int] :
? [N2: int] :
( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
& ( member_int @ N2 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_8036_infinite__int__iff__unbounded,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M3: int] :
? [N2: int] :
( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
& ( member_int @ N2 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_8037_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_8038_dvd__imp__le__int,axiom,
! [I2: int,D: int] :
( ( I2 != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).
% dvd_imp_le_int
thf(fact_8039_nat__abs__mult__distrib,axiom,
! [W2: int,Z: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_8040_abs__mod__less,axiom,
! [L: int,K2: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K2 @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_8041_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_8042_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_8043_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_8044_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K2: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_8045_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_8046_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_8047_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arctan_ubound
thf(fact_8048_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_8049_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arctan_bounded
thf(fact_8050_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_8051_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_8052_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_8053_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L2 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_8054_arctan__double,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_8055_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T6: real] :
( ( ord_less_eq_real @ zero_zero_real @ T6 )
=> ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T6 ) )
=> ( Y
!= ( sin_real @ T6 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_8056_sin__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( sin_real @ X )
= ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_8057_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_8058_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_8059_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_8060_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_8061_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
& ( ord_less_eq_real @ Y3 @ pi )
& ( ( sin_real @ Y3 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y3 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_8062_cos__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( cos_real @ X )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_8063_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_8064_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_8065_tan__total,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
& ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y6 )
= Y ) )
=> ( Y6 = X4 ) ) ) ).
% tan_total
thf(fact_8066_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_8067_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y @ X )
= ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_8068_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_8069_lemma__tan__total1,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_8070_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% cos_double_less_one
thf(fact_8071_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_8072_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_8073_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_8074_tan__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% tan_less_zero
thf(fact_8075_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_8076_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_8077_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% tan_bound_pi2
thf(fact_8078_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_8079_arctan,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_8080_arctan__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arctan @ ( tan_real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_8081_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( tan_real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_8082_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
& ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
& ( ( tan_real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_8083_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ T6 )
& ( ord_less_real @ T6 @ X )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_8084_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T6: real] :
( ( ord_less_real @ X @ T6 )
& ( ord_less_real @ T6 @ zero_zero_real )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_8085_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V1022390504157884413omplex @ Z )
= one_one_real )
=> ~ ! [T6: real] :
( ( ord_less_eq_real @ zero_zero_real @ T6 )
=> ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_8086_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22 = zero_zero_int )
=> ( A32
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q3 @ A22 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn_int @ R3 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_8087_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_8088_forall__pos__mono__1,axiom,
! [P2: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P2 @ D4 )
=> ( P2 @ E ) ) )
=> ( ! [N3: nat] : ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P2 @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_8089_forall__pos__mono,axiom,
! [P2: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P2 @ D4 )
=> ( P2 @ E ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P2 @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_8090_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_8091_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% ln_inverse
thf(fact_8092_log__inverse,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_8093_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_8094_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( inverse_inverse_real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_8095_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( ( K2
= ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_8096_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K2: int,Q2: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
=> ( ( K2
= ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_8097_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_power_real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_8098_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23 = zero_zero_int )
& ( A33
= ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
& ( L2 != zero_zero_int )
& ( K3
= ( times_times_int @ Q5 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q5 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L2 ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_8099_arctan__def,axiom,
( arctan
= ( ^ [Y5: real] :
( the_real
@ ^ [X5: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
& ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X5 )
= Y5 ) ) ) ) ) ).
% arctan_def
thf(fact_8100_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sinh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_ln_real
thf(fact_8101_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_8102_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_8103_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% sinh_real_neg_iff
thf(fact_8104_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% sinh_real_pos_iff
thf(fact_8105_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_8106_cosh__real__pos,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% cosh_real_pos
thf(fact_8107_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_8108_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_8109_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_8110_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_8111_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_8112_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X ) )
!= zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
thf(fact_8113_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( cosh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_ln_real
thf(fact_8114_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_8115_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_8116_horner__sum__of__bool__2__less,axiom,
! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_8117_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_8118_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_8119_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_8120_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_8121_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X5: real] :
( the_int
@ ^ [Z5: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X5 )
& ( ord_less_real @ X5 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_8122_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_8123_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M3: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_8124_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X5: rat] :
( the_int
@ ^ [Z5: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X5 )
& ( ord_less_rat @ X5 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_8125_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_8126_xor__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
!= ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% xor_negative_int_iff
thf(fact_8127_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_8128_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_8129_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_8130_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_8131_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_8132_abs__rat__def,axiom,
( abs_abs_rat
= ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_8133_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S3: rat] :
( ( ord_less_rat @ zero_zero_rat @ S3 )
=> ! [T6: rat] :
( ( ord_less_rat @ zero_zero_rat @ T6 )
=> ( R2
!= ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_8134_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_rat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% less_eq_rat_def
thf(fact_8135_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_8136_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_8137_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_8138_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M3: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_8139_normalize__negative,axiom,
! [Q2: int,P: int] :
( ( ord_less_int @ Q2 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P @ Q2 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_8140_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_8141_VEBT_Osize_I3_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size(3)
thf(fact_8142_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X9: nat > real] :
! [J3: nat] :
? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ M8 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M8 @ N2 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_8143_sum__power2,axiom,
! [K2: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_8144_or__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
| ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% or_negative_int_iff
thf(fact_8145_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_8146_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
= ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_8147_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P2 @ M3 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less_eq
thf(fact_8148_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P2 @ M3 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_8149_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_8150_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_8151_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_8152_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_8153_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_8154_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_8155_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_8156_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_8157_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P @ Q2 ) )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_8158_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K2: num] :
( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_8159_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_8160_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_8161_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_8162_VEBT_Osize__gen_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size_gen(1)
thf(fact_8163_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_8164_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_8165_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_8166_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size_gen(2)
thf(fact_8167_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_8168_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_8169_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_8170_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_8171_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_8172_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_8173_signed__take__bit__negative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ zero_zero_int )
= ( bit_se1146084159140164899it_int @ K2 @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_8174_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_8175_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_8176_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_8177_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_8178_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_8179_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_8180_card__less__Suc2,axiom,
! [M5: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M5 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M5 )
& ( ord_less_nat @ K3 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M5 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_8181_card__less__Suc,axiom,
! [M5: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M5 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M5 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M5 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_8182_card__less,axiom,
! [M5: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M5 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M5 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_8183_subset__card__intvl__is__intvl,axiom,
! [A4: set_nat,K2: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) )
=> ( A4
= ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_8184_bit__imp__take__bit__positive,axiom,
! [N: nat,M2: nat,K2: int] :
( ( ord_less_nat @ N @ M2 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ N )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_8185_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_8186_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_8187_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2 != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_8188_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_8189_int__bit__bound,axiom,
! [K2: int] :
~ ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ N3 @ M4 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ M4 )
= ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_8190_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_8191_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_8192_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_8193_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_8194_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M2 )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_8195_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_8196_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_8197_wmin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_8198_push__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_8199_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_8200_wmin__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_8201_bit__push__bit__iff__int,axiom,
! [M2: nat,K2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K2 ) @ N )
= ( ( ord_less_eq_nat @ M2 @ N )
& ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_8202_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q2 ) @ N )
= ( ( ord_less_eq_nat @ M2 @ N )
& ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_8203_binomial__def,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [K7: set_nat] :
( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
& ( ( finite_card_nat @ K7 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_8204_wmax__insertI,axiom,
! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ YS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_8205_smax__insertI,axiom,
! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ Y7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_8206_smin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_8207_wmax__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ X7 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_8208_smin__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] :
( ( X7 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_8209_smax__emptyI,axiom,
! [Y7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ Y7 )
=> ( ( Y7 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_8210_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_8211_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_8212_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_8213_Suc__0__div__numeral,axiom,
! [K2: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
= ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).
% Suc_0_div_numeral
thf(fact_8214_Suc__0__mod__numeral,axiom,
! [K2: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
= ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).
% Suc_0_mod_numeral
thf(fact_8215_finite__enumerate,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
& ! [N5: nat] :
( ( ord_less_nat @ N5 @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_8216_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_8217_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_8218_bezw_Osimps,axiom,
( bezw
= ( ^ [X5: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X5 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_8219_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_8220_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_8221_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P6 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_8222_drop__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% drop_bit_negative_int_iff
thf(fact_8223_gcd__pos__int,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M2 @ N ) )
= ( ( M2 != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
thf(fact_8224_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_8225_gcd__le1__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% gcd_le1_int
thf(fact_8226_gcd__le2__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% gcd_le2_int
thf(fact_8227_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_8228_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_8229_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_8230_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_8231_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
% gcd_0_nat
thf(fact_8232_gcd__nat_Oright__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ zero_zero_nat )
= A ) ).
% gcd_nat.right_neutral
thf(fact_8233_gcd__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( gcd_gcd_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_8234_gcd__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ A )
= A ) ).
% gcd_nat.left_neutral
thf(fact_8235_gcd__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_8236_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_8237_gcd__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
= ( ( M2 != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_8238_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_8239_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_8240_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_8241_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_8242_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_8243_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_8244_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_8245_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_8246_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_8247_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_8248_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_8249_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_8250_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_8251_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_8252_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_8253_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_8254_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_8255_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_8256_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% gcd_le2_nat
thf(fact_8257_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% gcd_le1_nat
thf(fact_8258_gcd__diff1__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M2 @ N ) @ N )
= ( gcd_gcd_nat @ M2 @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_8259_gcd__diff2__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M2 ) @ N )
= ( gcd_gcd_nat @ M2 @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_8260_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_8261_gcd__nat_Osimps,axiom,
( gcd_gcd_nat
= ( ^ [X5: nat,Y5: nat] : ( if_nat @ ( Y5 = zero_zero_nat ) @ X5 @ ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_8262_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_8263_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_8264_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_8265_real__root__power,axiom,
! [N: nat,X: real,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K2 ) )
= ( power_power_real @ ( root @ N @ X ) @ K2 ) ) ) ).
% real_root_power
thf(fact_8266_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_8267_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X4: nat,Y3: nat] :
( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_8268_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X4: nat,Y3: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_8269_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( sgn_sgn_real @ ( root @ N @ X ) )
= ( sgn_sgn_real @ X ) ) ) ).
% sgn_root
thf(fact_8270_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_8271_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_8272_real__root__strict__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_8273_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_8274_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_8275_real__root__strict__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_8276_real__root__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_8277_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_8278_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_8279_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_8280_real__root__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_8281_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_8282_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_8283_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_8284_log__root,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_8285_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_8286_split__root,axiom,
! [P2: real > $o,N: nat,X: real] :
( ( P2 @ ( root @ N @ X ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N )
=> ! [Y5: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
= X )
=> ( P2 @ Y5 ) ) ) ) ) ).
% split_root
thf(fact_8287_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_8288_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X5: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X5 )
@ ^ [X5: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X5 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_8289_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_8290_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_8291_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_8292_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_8293_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M3: nat,N2: nat] :
( ( dvd_dvd_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_8294_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_8295_Inf__nat__def1,axiom,
! [K5: set_nat] :
( ( K5 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_8296_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X5 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X5 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_8297_int_Oabs__induct,axiom,
! [P2: int > $o,X: int] :
( ! [Y3: product_prod_nat_nat] : ( P2 @ ( abs_Integ @ Y3 ) )
=> ( P2 @ X ) ) ).
% int.abs_induct
thf(fact_8298_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X4: nat,Y3: nat] :
( Z
!= ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_8299_nat_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% nat.abs_eq
thf(fact_8300_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_8301_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_8302_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X5: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X5 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_8303_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_8304_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_8305_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_8306_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_8307_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_8308_max__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_8309_min__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_8310_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_8311_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X5: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_8312_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% prod_encode_def
thf(fact_8313_Gcd__remove0__nat,axiom,
! [M5: set_nat] :
( ( finite_finite_nat @ M5 )
=> ( ( gcd_Gcd_nat @ M5 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_8314_prod__encode__eq,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y ) )
= ( X = Y ) ) ).
% prod_encode_eq
thf(fact_8315_Gcd__in,axiom,
! [A4: set_nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( member_nat @ B3 @ A4 )
=> ( member_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A4 ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A4 ) @ A4 ) ) ) ).
% Gcd_in
thf(fact_8316_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_8317_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_8318_nat_Orep__eq,axiom,
( nat2
= ( ^ [X5: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X5 ) ) ) ) ).
% nat.rep_eq
thf(fact_8319_prod__encode__prod__decode__aux,axiom,
! [K2: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
= ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_8320_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X5: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X5 ) ) ) ) ).
% uminus_int_def
thf(fact_8321_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X5 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X5 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_8322_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_8323_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_8324_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C2 @ Y )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less_nat @ C2 @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C2 )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_8325_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_8326_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_8327_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_8328_less__rat,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% less_rat
thf(fact_8329_zero__notin__Suc__image,axiom,
! [A4: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_8330_Rat__induct__pos,axiom,
! [P2: rat > $o,Q2: rat] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( P2 @ ( fract @ A3 @ B3 ) ) )
=> ( P2 @ Q2 ) ) ).
% Rat_induct_pos
thf(fact_8331_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_8332_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_8333_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_8334_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_8335_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_8336_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_8337_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% Fract_less_zero_iff
thf(fact_8338_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% zero_less_Fract_iff
thf(fact_8339_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% one_less_Fract_iff
thf(fact_8340_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_int @ A @ B ) ) ) ).
% Fract_less_one_iff
thf(fact_8341_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_Fract_iff
thf(fact_8342_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% Fract_le_zero_iff
thf(fact_8343_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% Fract_le_one_iff
thf(fact_8344_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% one_le_Fract_iff
thf(fact_8345_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_8346_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
thf(fact_8347_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_8348_less__eq__int__def,axiom,
( ord_less_eq_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_8349_finite__int__iff__bounded__le,axiom,
( finite_finite_int
= ( ^ [S7: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_8350_finite__int__iff__bounded,axiom,
( finite_finite_int
= ( ^ [S7: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_8351_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_8352_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_8353_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_8354_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X5: int] : ( plus_plus_int @ X5 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_8355_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_8356_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_8357_range__mult,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
thf(fact_8358_surj__prod__encode,axiom,
( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
= top_top_set_nat ) ).
% surj_prod_encode
thf(fact_8359_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_8360_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_8361_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_8362_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_8363_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_8364_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_8365_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_8366_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_8367_root__def,axiom,
( root
= ( ^ [N2: nat,X5: real] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N2 ) )
@ X5 ) ) ) ) ).
% root_def
thf(fact_8368_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_8369_has__real__derivative__neg__dec__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_8370_has__real__derivative__pos__inc__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_8371_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_8372_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_8373_DERIV__isconst3,axiom,
! [A: real,B: real,X: real,Y: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_8374_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_8375_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_8376_DERIV__pos__inc__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_8377_DERIV__neg__dec__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_8378_DERIV__neg__dec__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_8379_DERIV__pos__inc__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_8380_MVT2,axiom,
! [A: real,B: real,F: real > real,F6: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F6 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_8381_DERIV__local__const,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
=> ( ( F @ X )
= ( F @ Y3 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
thf(fact_8382_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln
thf(fact_8383_DERIV__local__max,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
thf(fact_8384_DERIV__local__min,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
thf(fact_8385_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln_divide
thf(fact_8386_DERIV__pow,axiom,
! [N: nat,X: real,S: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( power_power_real @ X5 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% DERIV_pow
thf(fact_8387_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( has_fi5821293074295781190e_real
@ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
@ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_8388_DERIV__log,axiom,
! [X: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_8389_DERIV__fun__powr,axiom,
! [G: real > real,M2: real,X: real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( powr_real @ ( G @ X5 ) @ R2 )
@ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_8390_DERIV__powr,axiom,
! [G: real > real,M2: real,X: real,F: real > real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( powr_real @ ( G @ X5 ) @ ( F @ X5 ) )
@ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M2 @ ( F @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_powr
thf(fact_8391_arcosh__real__has__field__derivative,axiom,
! [X: real,A4: set_real] :
( ( ord_less_real @ one_one_real @ X )
=> ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_8392_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_real_sqrt
thf(fact_8393_DERIV__real__sqrt__generic,axiom,
! [X: real,D6: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D6
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D6
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_8394_artanh__real__has__field__derivative,axiom,
! [X: real,A4: set_real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_8395_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X4 @ N2 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] :
( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X5 @ ( suc @ N2 ) ) ) )
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_8396_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_8397_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arcsin
thf(fact_8398_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T6: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_8399_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [T6: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_8400_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( X != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_8401_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ T6 )
& ( ord_less_real @ T6 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_8402_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_8403_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H2 @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ H2 @ T6 )
& ( ord_less_eq_real @ T6 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ? [T6: real] :
( ( ord_less_real @ H2 @ T6 )
& ( ord_less_real @ T6 @ zero_zero_real )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_8404_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T6: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
& ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_8405_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ? [T6: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_8406_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T6 )
& ( ord_less_eq_real @ T6 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ? [T6: real] :
( ( ord_less_real @ A @ T6 )
& ( ord_less_real @ T6 @ C2 )
& ( ( F @ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C2 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_8407_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T6 )
& ( ord_less_eq_real @ T6 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_real @ C2 @ B )
=> ? [T6: real] :
( ( ord_less_real @ C2 @ T6 )
& ( ord_less_real @ T6 @ B )
& ( ( F @ B )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C2 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_8408_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T6 )
& ( ord_less_eq_real @ T6 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( X != C2 )
=> ? [T6: real] :
( ( ( ord_less_real @ X @ C2 )
=> ( ( ord_less_real @ X @ T6 )
& ( ord_less_real @ T6 @ C2 ) ) )
& ( ~ ( ord_less_real @ X @ C2 )
=> ( ( ord_less_real @ C2 @ T6 )
& ( ord_less_real @ T6 @ X ) ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C2 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ M3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_8409_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K2: nat,B5: real] :
( ! [M: nat,T6: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M4: nat,T7: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T7 )
& ( ord_less_eq_real @ T7 @ H2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U2: real] :
( minus_minus_real @ ( Diff @ M4 @ U2 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M4 ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M4 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T7 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_8410_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X10: real] :
( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_8411_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D6: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D6
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D6
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( D6
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_8412_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arccos
thf(fact_8413_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_8414_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_8415_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_8416_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( sin_real @ ( arccos @ X ) )
!= zero_zero_real ) ) ) ).
% sin_arccos_nonzero
thf(fact_8417_Gcd__eq__Max,axiom,
! [M5: set_nat] :
( ( finite_finite_nat @ M5 )
=> ( ( M5 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M5 )
=> ( ( gcd_Gcd_nat @ M5 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M3: nat] :
( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M3 ) )
@ M5 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_8418_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_8419_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X3: real] :
( ( ( X3 != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X3 ) ) @ R3 ) )
=> ( ord_less_real @ ( F @ X3 ) @ zero_zero_real ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_8420_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( L != zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X3: real] :
( ( ( X3 != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X3 ) ) @ R3 ) )
=> ( ( F @ X3 )
!= zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_8421_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X3: real] :
( ( ( X3 != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X3 ) ) @ R3 ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_8422_card__le__Suc__Max,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_8423_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X9: set_nat] : ( if_nat @ ( X9 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_8424_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M3: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M3 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_8425_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M2 @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D5: nat] :
( ( dvd_dvd_nat @ D5 @ M2 )
& ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_8426_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N5: nat] :
( member_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_8427_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N5: nat] :
( member_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_8428_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_8429_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
thf(fact_8430_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( filterlim_nat_nat @ ( times_times_nat @ C2 ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_8431_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( filterlim_nat_nat
@ ^ [X5: nat] : ( times_times_nat @ X5 @ C2 )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_8432_LIMSEQ__inverse__zero,axiom,
! [X7: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_real @ R3 @ ( X7 @ N3 ) ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( X7 @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_8433_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( root @ N2 @ C2 )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_8434_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_8435_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_8436_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_8437_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C2 ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_8438_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ C2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_8439_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_8440_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_8441_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L4: real] :
( ! [N5: nat] :
( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
@ L4 )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat )
& ! [N5: nat] :
( ord_less_eq_real @ L4
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_8442_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_8443_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_8444_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ B @ X4 )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_8445_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F3: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X5: real] : ( power_power_real @ ( F @ X5 ) @ N )
@ at_top_real
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_8446_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F3: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X5: real] : ( power_power_real @ ( F @ X5 ) @ N )
@ at_bot_real
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_8447_eventually__sequentially__seg,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually_nat
@ ^ [N2: nat] : ( P2 @ ( plus_plus_nat @ N2 @ K2 ) )
@ at_top_nat )
= ( eventually_nat @ P2 @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_8448_eventually__sequentiallyI,axiom,
! [C2: nat,P2: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq_nat @ C2 @ X4 )
=> ( P2 @ X4 ) )
=> ( eventually_nat @ P2 @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_8449_eventually__sequentially,axiom,
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ at_top_nat )
= ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( P2 @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_8450_le__sequentially,axiom,
! [F3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
= ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F3 ) ) ) ).
% le_sequentially
thf(fact_8451_sequentially__offset,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually_nat @ P2 @ at_top_nat )
=> ( eventually_nat
@ ^ [I4: nat] : ( P2 @ ( plus_plus_nat @ I4 @ K2 ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_8452_eventually__at__left__real,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( eventually_real
@ ^ [X5: real] : ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B @ A ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% eventually_at_left_real
thf(fact_8453_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_8454_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_8455_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_8456_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_8457_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B: nat] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_8458_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_8459_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_8460_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8461_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_8462_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_8463_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_8464_isCont__inverse__function2,axiom,
! [A: real,X: real,B: real,G: real > real,F: real > real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_8465_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% isCont_arcosh
thf(fact_8466_DERIV__inverse__function,axiom,
! [F: real > real,D6: real,G: real > real,X: real,A: real,B: real] :
( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
=> ( ( D6 != zero_zero_real )
=> ( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y3: real] :
( ( ord_less_real @ A @ Y3 )
=> ( ( ord_less_real @ Y3 @ B )
=> ( ( F @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_8467_isCont__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_8468_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_8469_LIM__less__bound,axiom,
! [B: real,X: real,F: real > real] :
( ( ord_less_real @ B @ X )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_8470_isCont__artanh,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% isCont_artanh
thf(fact_8471_isCont__inverse__function,axiom,
! [D: real,X: real,G: real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_8472_GMVT_H,axiom,
! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F6: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ? [C: real] :
( ( ord_less_real @ A @ C )
& ( ord_less_real @ C @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F6 @ C ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_8473_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_8474_eventually__at__right__real,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( eventually_real
@ ^ [X5: real] : ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% eventually_at_right_real
thf(fact_8475_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_8476_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_8477_GMVT,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [G_c: real,F_c: real,C: real] :
( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
& ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
& ( ord_less_real @ A @ C )
& ( ord_less_real @ C @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_8478_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_8479_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( set_or1210151606488870762an_nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_8480_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_8481_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K2 ) @ ( insert_nat @ K2 @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_8482_MVT,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_8483_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
thf(fact_8484_Rolle__deriv,axiom,
! [A: real,B: real,F: real > real,F6: real > real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( F6 @ Z3 )
= ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_8485_mvt,axiom,
! [A: real,B: real,F: real > real,F6: real > real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less_real @ A @ Xi )
=> ( ( ord_less_real @ Xi @ B )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
!= ( F6 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_8486_DERIV__isconst__end,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( F @ B )
= ( F @ A ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_8487_DERIV__neg__imp__decreasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_8488_DERIV__pos__imp__increasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_8489_DERIV__isconst2,axiom,
! [A: real,B: real,F: real > real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( F @ X )
= ( F @ A ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_8490_Rolle,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% Rolle
thf(fact_8491_uniformity__complex__def,axiom,
( topolo896644834953643431omplex
= ( comple8358262395181532106omplex
@ ( image_5971271580939081552omplex
@ ^ [E3: real] :
( princi3496590319149328850omplex
@ ( collec8663557070575231912omplex
@ ( produc6771430404735790350plex_o
@ ^ [X5: complex,Y5: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X5 @ Y5 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_complex_def
thf(fact_8492_uniformity__real__def,axiom,
( topolo1511823702728130853y_real
= ( comple2936214249959783750l_real
@ ( image_2178119161166701260l_real
@ ^ [E3: real] :
( princi6114159922880469582l_real
@ ( collec3799799289383736868l_real
@ ( produc5414030515140494994real_o
@ ^ [X5: real,Y5: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X5 @ Y5 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_real_def
thf(fact_8493_eventually__prod__sequentially,axiom,
! [P2: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P2 @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N4: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ N4 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ N4 @ N2 )
=> ( P2 @ ( product_Pair_nat_nat @ N2 @ M3 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_8494_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_8495_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_8496_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_8497_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( order_mono_nat_nat
@ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M3 ) @ M3 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_8498_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_8499_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_8500_surj__int__encode,axiom,
( ( image_int_nat @ nat_int_encode @ top_top_set_int )
= top_top_set_nat ) ).
% surj_int_encode
thf(fact_8501_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_8502_int__encode__eq,axiom,
! [X: int,Y: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y ) )
= ( X = Y ) ) ).
% int_encode_eq
thf(fact_8503_log__inj,axiom,
! [B: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% log_inj
thf(fact_8504_measure__function__int,axiom,
fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% measure_function_int
thf(fact_8505_inj__prod__encode,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A4 ) ).
% inj_prod_encode
thf(fact_8506_inj__int__encode,axiom,
! [A4: set_int] : ( inj_on_int_nat @ nat_int_encode @ A4 ) ).
% inj_int_encode
thf(fact_8507_inj__Suc,axiom,
! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_8508_inj__on__diff__nat,axiom,
! [N6: set_nat,K2: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N6 )
=> ( ord_less_eq_nat @ K2 @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K2 )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_8509_inj__on__set__encode,axiom,
inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% inj_on_set_encode
thf(fact_8510_surj__int__decode,axiom,
( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
= top_top_set_int ) ).
% surj_int_decode
thf(fact_8511_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_8512_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_8513_inj__int__decode,axiom,
! [A4: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A4 ) ).
% inj_int_decode
thf(fact_8514_int__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y ) )
= ( X = Y ) ) ).
% int_decode_eq
thf(fact_8515_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( X != zero_zero_real )
| ( ord_less_int @ zero_zero_int @ N ) )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_int_real @ X @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_8516_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_8517_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_8518_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_8519_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_8520_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_8521_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_8522_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_8523_take__bit__num__simps_I1_J,axiom,
! [M2: num] :
( ( bit_take_bit_num @ zero_zero_nat @ M2 )
= none_num ) ).
% take_bit_num_simps(1)
thf(fact_8524_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I4: int,N2: nat] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( N2 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_8525_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X4 )
& ( ord_less_real @ X4 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_8526_Rats__no__bot__less,axiom,
! [X: real] :
? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_real @ X4 @ X ) ) ).
% Rats_no_bot_less
thf(fact_8527_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_8528_min__weak__def,axiom,
( fun_min_weak
= ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% min_weak_def
thf(fact_8529_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_8530_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
thf(fact_8531_min__strict__def,axiom,
( fun_min_strict
= ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_8532_max__weak__def,axiom,
( fun_max_weak
= ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% max_weak_def
thf(fact_8533_max__strict__def,axiom,
( fun_max_strict
= ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_8534_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M3: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_8535_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_8536_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_8537_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_8538_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_8539_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_8540_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_8541_pos__deriv__imp__strict__mono,axiom,
! [F: real > real,F6: real > real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X4 ) )
=> ( order_7092887310737990675l_real @ F ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_8542_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_8543_infinite__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( order_5726023648592871131at_nat @ R3 )
& ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_8544_positive__rat,axiom,
! [A: int,B: int] :
( ( positive @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% positive_rat
thf(fact_8545_less__rat__def,axiom,
( ord_less_rat
= ( ^ [X5: rat,Y5: rat] : ( positive @ ( minus_minus_rat @ Y5 @ X5 ) ) ) ) ).
% less_rat_def
thf(fact_8546_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X5: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X5 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X5 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_8547_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X5: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_8548_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_8549_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_8550_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_8551_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_8552_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M2 @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_8553_min__diff,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I2 ) ) ).
% min_diff
thf(fact_8554_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M2 )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_8555_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_8556_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
= ( ord_less_nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_8557_inf__int__def,axiom,
inf_inf_int = ord_min_int ).
% inf_int_def
thf(fact_8558_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_8559_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod_nat_nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_8560_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_8561_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_8562_sorted__list__of__set__lessThan__Suc,axiom,
! [K2: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K2 ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K2 ) ) @ ( cons_nat @ K2 @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_8563_sorted__list__of__set__atMost__Suc,axiom,
! [K2: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K2 ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K2 ) ) @ ( cons_nat @ ( suc @ K2 ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_8564_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X4: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X4 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_8565_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X4 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_8566_inj__list__encode,axiom,
! [A4: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A4 ) ).
% inj_list_encode
thf(fact_8567_surj__list__encode,axiom,
( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
= top_top_set_nat ) ).
% surj_list_encode
thf(fact_8568_list__encode__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y ) )
= ( X = Y ) ) ).
% list_encode_eq
thf(fact_8569_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ nil_nat )
= zero_zero_nat ) ).
% list_encode.simps(1)
thf(fact_8570_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_8571_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_8572_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_8573_upto_Opsimps,axiom,
! [I2: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
=> ( ( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_8574_upto__Nil,axiom,
! [I2: int,J: int] :
( ( ( upto @ I2 @ J )
= nil_int )
= ( ord_less_int @ J @ I2 ) ) ).
% upto_Nil
thf(fact_8575_upto__Nil2,axiom,
! [I2: int,J: int] :
( ( nil_int
= ( upto @ I2 @ J ) )
= ( ord_less_int @ J @ I2 ) ) ).
% upto_Nil2
thf(fact_8576_upto__empty,axiom,
! [J: int,I2: int] :
( ( ord_less_int @ J @ I2 )
=> ( ( upto @ I2 @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_8577_upto__single,axiom,
! [I2: int] :
( ( upto @ I2 @ I2 )
= ( cons_int @ I2 @ nil_int ) ) ).
% upto_single
thf(fact_8578_nth__upto,axiom,
! [I2: int,K2: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ J )
=> ( ( nth_int @ ( upto @ I2 @ J ) @ K2 )
= ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).
% nth_upto
thf(fact_8579_length__upto,axiom,
! [I2: int,J: int] :
( ( size_size_list_int @ ( upto @ I2 @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_8580_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_8581_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_8582_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_8583_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_8584_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_8585_distinct__upto,axiom,
! [I2: int,J: int] : ( distinct_int @ ( upto @ I2 @ J ) ) ).
% distinct_upto
thf(fact_8586_upto__aux__def,axiom,
( upto_aux
= ( ^ [I4: int,J3: int] : ( append_int @ ( upto @ I4 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_8587_upto__code,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( upto_aux @ I4 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_8588_upto__split2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ).
% upto_split2
thf(fact_8589_upto__split1,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K2 ) ) ) ) ) ).
% upto_split1
thf(fact_8590_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_8591_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_8592_upto__rec1,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_8593_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list_int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_8594_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_8595_upto__rec2,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_8596_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_8597_upto__split3,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ) ).
% upto_split3
thf(fact_8598_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X4 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_8599_remdups__upt,axiom,
! [M2: nat,N: nat] :
( ( remdups_nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% remdups_upt
thf(fact_8600_hd__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( hd_nat @ ( upt @ I2 @ J ) )
= I2 ) ) ).
% hd_upt
thf(fact_8601_drop__upt,axiom,
! [M2: nat,I2: nat,J: nat] :
( ( drop_nat @ M2 @ ( upt @ I2 @ J ) )
= ( upt @ ( plus_plus_nat @ I2 @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_8602_length__upt,axiom,
! [I2: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ I2 ) ) ).
% length_upt
thf(fact_8603_take__upt,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M2 ) @ N )
=> ( ( take_nat @ M2 @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus_nat @ I2 @ M2 ) ) ) ) ).
% take_upt
thf(fact_8604_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( upt @ I2 @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_8605_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sorted_list_of_set_range
thf(fact_8606_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( ( upt @ I2 @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_8607_nth__upt,axiom,
! [I2: nat,K2: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J )
=> ( ( nth_nat @ ( upt @ I2 @ J ) @ K2 )
= ( plus_plus_nat @ I2 @ K2 ) ) ) ).
% nth_upt
thf(fact_8608_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_8609_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_8610_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M2 @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_8611_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_8612_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M2 ) )
= ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_8613_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_8614_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_8615_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_8616_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_8617_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_8618_distinct__upt,axiom,
! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).
% distinct_upt
thf(fact_8619_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_8620_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( upt @ I2 @ J )
= ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_8621_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_8622_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( plus_plus_nat @ J @ K2 ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_8623_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I2 @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I2 @ J )
& ( I2 = X )
& ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_8624_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_8625_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_8626_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_8627_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).
% sum_list_upt
thf(fact_8628_card__length__sum__list__rec,axiom,
! [M2: nat,N6: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M2 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( groups4561878855575611511st_nat @ L2 )
= N6 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M2 @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N6 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_8629_card__length__sum__list,axiom,
! [M2: nat,N6: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( groups4561878855575611511st_nat @ L2 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M2 ) @ one_one_nat ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_8630_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_8631_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_8632_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I2: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_8633_sorted__upto,axiom,
! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).
% sorted_upto
thf(fact_8634_sorted__wrt__upto,axiom,
! [I2: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J ) ) ).
% sorted_wrt_upto
thf(fact_8635_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl_nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_8636_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M2 ) ) )
= ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
@ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_8637_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_8638_vimage__Suc__insert__0,axiom,
! [A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A4 ) )
= ( vimage_nat_nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_8639_vimage__Suc__insert__Suc,axiom,
! [N: nat,A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A4 ) )
= ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_8640_finite__vimage__Suc__iff,axiom,
! [F3: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
= ( finite_finite_nat @ F3 ) ) ).
% finite_vimage_Suc_iff
thf(fact_8641_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_8642_set__encode__vimage__Suc,axiom,
! [A4: set_nat] :
( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A4 ) )
= ( divide_divide_nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_8643_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) )
@ ^ [Uu3: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_8644_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% natLeq_def
thf(fact_8645_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_8646_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_8647_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_8648_total__pair__less,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A4 @ fun_pair_less ) ).
% total_pair_less
thf(fact_8649_Arg__correct,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( ( sgn_sgn_complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_8650_not__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% not_negative_int_iff
thf(fact_8651_not__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K2 ) )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% not_nonnegative_int_iff
thf(fact_8652_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn_complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_8653_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_complex
@ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
@ ( set_ord_lessThan_nat @ N )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_8654_bij__betw__nth__root__unity,axiom,
! [C2: complex,N: nat] :
( ( C2 != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C2 ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_8655_Arg__def,axiom,
( arg
= ( ^ [Z5: complex] :
( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A5: real] :
( ( ( sgn_sgn_complex @ Z5 )
= ( cis @ A5 ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
& ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_8656_bij__betw__Suc,axiom,
! [M5: set_nat,N6: set_nat] :
( ( bij_betw_nat_nat @ suc @ M5 @ N6 )
= ( ( image_nat_nat @ suc @ M5 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_8657_bij__int__decode,axiom,
bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% bij_int_decode
thf(fact_8658_bij__int__encode,axiom,
bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% bij_int_encode
thf(fact_8659_bij__list__encode,axiom,
bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% bij_list_encode
thf(fact_8660_bij__prod__encode,axiom,
bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% bij_prod_encode
thf(fact_8661_Gcd__int__set__eq__fold,axiom,
! [Xs: list_int] :
( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
= ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).
% Gcd_int_set_eq_fold
thf(fact_8662_Gcd__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_8663_sort__upt,axiom,
! [M2: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X5: nat] : X5
@ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sort_upt
thf(fact_8664_sort__upto,axiom,
! [I2: int,J: int] :
( ( linord1735203802627413978nt_int
@ ^ [X5: int] : X5
@ ( upto @ I2 @ J ) )
= ( upto @ I2 @ J ) ) ).
% sort_upto
thf(fact_8665_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) )
= ( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_8666_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).
% pair_leq_def
thf(fact_8667_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_8668_wf__pair__less,axiom,
wf_Pro7803398752247294826at_nat @ fun_pair_less ).
% wf_pair_less
thf(fact_8669_wf__int__ge__less__than2,axiom,
! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).
% wf_int_ge_less_than2
thf(fact_8670_wf__int__ge__less__than,axiom,
! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).
% wf_int_ge_less_than
thf(fact_8671_cauchyD,axiom,
! [X7: nat > rat,R2: rat] :
( ( cauchy @ X7 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ K @ N5 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M4 ) @ ( X7 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_8672_cauchyI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [M: nat] :
( ( ord_less_eq_nat @ K4 @ M )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M ) @ ( X7 @ N3 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X7 ) ) ).
% cauchyI
thf(fact_8673_cauchy__imp__bounded,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ? [B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ B3 )
& ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N5 ) ) @ B3 ) ) ) ).
% cauchy_imp_bounded
thf(fact_8674_cauchy__def,axiom,
( cauchy
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ K3 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_8675_le__Real,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_8676_cauchy__not__vanishes,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ B3 )
& ? [K: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ K @ N5 )
=> ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X7 @ N5 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_8677_vanishes__mult__bounded,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ? [A8: rat] :
( ( ord_less_rat @ zero_zero_rat @ A8 )
& ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N3 ) ) @ A8 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_8678_vanishes__def,axiom,
( vanishes
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_8679_vanishesI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X7 ) ) ).
% vanishesI
thf(fact_8680_vanishesD,axiom,
! [X7: nat > rat,R2: rat] :
( ( vanishes @ X7 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K: nat] :
! [N5: nat] :
( ( ord_less_eq_nat @ K @ N5 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N5 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_8681_cauchy__not__vanishes__cases,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B3: rat] :
( ( ord_less_rat @ zero_zero_rat @ B3 )
& ? [K: nat] :
( ! [N5: nat] :
( ( ord_less_eq_nat @ K @ N5 )
=> ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X7 @ N5 ) ) ) )
| ! [N5: nat] :
( ( ord_less_eq_nat @ K @ N5 )
=> ( ord_less_rat @ B3 @ ( X7 @ N5 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_8682_not__positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X7 @ N2 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_8683_positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( positive2 @ ( real2 @ X7 ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_8684_less__real__def,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] : ( positive2 @ ( minus_minus_real @ Y5 @ X5 ) ) ) ) ).
% less_real_def
thf(fact_8685_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X5: real] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( rep_real @ X5 @ N2 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_8686_Real_Opositive__def,axiom,
( positive2
= ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X9 @ N2 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_8687_le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_8688_enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S @ S2 )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S2 @ N3 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_8689_strict__mono__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).
% strict_mono_enumerate
thf(fact_8690_range__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
= S2 ) ) ).
% range_enumerate
thf(fact_8691_finite__le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_8692_bij__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat @ S2 ) ) ).
% bij_enumerate
thf(fact_8693_Least__eq__0,axiom,
! [P2: nat > $o] :
( ( P2 @ zero_zero_nat )
=> ( ( ord_Least_nat @ P2 )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_8694_Least__Suc,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ( ( ord_Least_nat @ P2 )
= ( suc
@ ( ord_Least_nat
@ ^ [M3: nat] : ( P2 @ ( suc @ M3 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_8695_Least__Suc2,axiom,
! [P2: nat > $o,N: nat,Q: nat > $o,M2: nat] :
( ( P2 @ N )
=> ( ( Q @ M2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ( ! [K: nat] :
( ( P2 @ ( suc @ K ) )
= ( Q @ K ) )
=> ( ( ord_Least_nat @ P2 )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
% Helper facts (32)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P2: real > $o] :
( ( P2 @ ( fChoice_real @ P2 ) )
= ( ? [X9: real] : ( P2 @ X9 ) ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya )
| ( xa = ya ) ) ).
%------------------------------------------------------------------------------